Roquette P.-Contributions to the History of Number Theory in the 20th Century(2013).pdf

Peter Roquette, Oberwolfach, March 2006

Peter Roquette

Contributionsto the History ofNumber Theory

in the 20th Century

Author:

Peter RoquetteRuprecht-Karls-Universität HeidelbergMathematisches InstitutIm Neuenheimer Feld 28869120 HeidelbergGermany

E-mail: [email protected]

2010 Mathematics Subject Classification (primary; secondary): 01-02, 03-03, 11-03, 12-03 , 16-03,20-03; 01A60, 01A70, 01A75, 11E04, 11E88, 11R18 11R37, 11U10

ISBN 978-3-03719-113-2

The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.

This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad-casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of usepermission of the copyright owner must be obtained.

© 2013 European Mathematical Society

Contact address:

European Mathematical Society Publishing HouseSeminar for Applied MathematicsETH-Zentrum SEW A27CH-8092 ZürichSwitzerland

Phone: +41 (0)44 632 34 36Email: [email protected]: www.ems-ph.org

Typeset using the author’s TEX files: I. Zimmermann, FreiburgPrinting and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany∞ Printed on acid free paper9 8 7 6 5 4 3 2 1

To my friend Günther Frei who introduced me to and kindled

my interest in the history of number theory

Preface

This volume contains my articles on the history of number theory except those whichare already included in my “Collected Papers”. All articles have been checked andreedited. Two articles which originally were written in German have been translated.I would like to thank all the people who have helped me to prepare this volume,foremost Keith Conrad and also Milena Hering who have streamlined my translation.

Particular thanks go to my wife Erika who has meticulously read and correctedthe whole manuscript.

November 2012 Peter Roquette

Contents

Preface vii

List of figures xi

1 The Brauer–Hasse–Noether Theorem 1

2 The remarkable career of Otto Grün 77

3 At Emmy Noether’s funeral 117

4 Emmy Noether and Hermann Weyl 129

5 Emmy Noether: The testimonials 163

6 Abraham Robinson and his infinitesimals 175

7 Cahit Arf and his invariant 189

8 Hasse–Arf–Langlands 223

9 Ernst Steinitz and abstract field theory 227

10 Heinrich-Wolfgang Leopoldt 239

11 On Hoechsmann’s Theorem 245

Acknowledgements 255

Bibliography 257

Name Index 273

Subject Index 277

List of Figures

Richard Brauer, Helmut Hasse, Emmy Noether 2

Otto Grün 79

Noether’s tomb 125

Emmy Noether, Hermann Weyl 146

Emmy Noether 1933 165

Abraham Robinson 176

Cahit Arf 190

Helmut Hasse, Cahit Arf, Robert Langlands 225

Ernst Steinitz 227

Heinrich-Wolfgang Leopoldt 239

Chapter 1

The Brauer–Hasse–Noether Theorem

Revised version of:

The Brauer–Hasse–Noether theorem in historical perspective.

Schriften der Math.-Phys. Klasse der Heidelberger Akademie der Wissenschaften Nr. 15 (2005).

1.1 Introduction 11.2 The Main Theorem: cyclic algebras 31.3 The paper: dedication to Hensel 61.4 The Local-Global Principle 111.5 From the LGP to the Main Theorem 201.6 The Brauer group and class field theory 311.7 The team: Noether, Brauer and Hasse 451.8 The American connection: Albert 561.9 Epilogue: Käte Hey 73

1.1 Introduction

The legacy of Helmut Hasse, consisting of letters, manuscripts and other papers, iskept at the Handschriftenabteilung of the University Library at Göttingen. Hasse hadan extensive correspondence; he liked to exchange mathematical ideas, results andmethods freely with his colleagues. There are more than 8 000 documents preserved.Although not all of them are of equal mathematical interest, searching through thistreasure can help us to assess the development of Number Theory through the 1920sand 1930s. Unfortunately, most of the correspondence is preserved on one side only,i.e., the letters sent to Hasse are available whereas many of the letters which had beensent from him, often handwritten, seem to be lost. So we have to interpolate, as far aspossible, from the replies to Hasse and from other contexts, in order to find out whathe had written in his outgoing letters.1

1An exception is the correspondence between Hasse and Richard Brauer. Thanks to Prof. Fred Brauer, theletters from Hasse to Richard Brauer are now available in Göttingen too.

2 1 The Brauer–Hasse–Noether Theorem

The present article is largely based on the letters and other documents which Ihave found concerning the

Brauer–Hasse–Noether Theorem

in the theory of algebras; this covers the years around 1931. Besides the documentsfrom the Hasse and the Brauer legacy in Göttingen, I shall also use some letters fromEmmy Noether to Richard Brauer which are preserved at the Bryn Mawr CollegeLibrary (Pennsylvania, USA).

We should be aware that the Brauer–Hasse–Noether Theorem, although to berated as a highlight, does not constitute the summit and end point of a development.We have to regard it as a step, important but not final, in a development which leadsto the view of class field theory as we see it today. By concentrating on the Brauer–Hasse–Noether Theorem we get only what may be called a snapshot within the greatedifice of class field theory.

A snapshot is not a panoramic view. Accordingly, the reader might miss severalaspects which also could throw some light on the position of the Brauer–Hasse–Noether theorem, its sources and its consequences, not only within algebraic numbertheory but also in other mathematical disciplines. It would have been impossible toinclude all these into this paper. Thus I have decided to present it as it is now, beingaware of its shortcomings with respect to the range of topics treated, as well as thetime span taken into consideration.

R. Brauer H. Hasse E. Noether

A preliminary version of this article had been written in connection with mylecture at the conference March 22–24, 2001 in Stuttgart which was dedicated tothe memory of Richard Brauer on the occasion of his 100th birthday. For Brauer,the cooperation with Noether and Hasse in this project constituted an unforgettable,exciting experience. Let us cite from a letter he wrote many years later, on March 3,1961, to Helmut Hasse:

… ist es 35 Jahre her, daß ich durch Sie mit der Klassenkörpertheoriebekannt geworden bin. Daß ich in Zusammenarbeit mit Ihnen und Emmy


Noether ein wenig dazu beitragen konnte, ist auch mir eine der schönstenErinnerungen, und ich werde die Aufregung der Tage, in denen die Arbeitentstand, nie vergessen.

… it is now 35 years since you introduced me to class field theory. Itbelongs to my most delightful memories that I was able, in cooperationwith you and Emmy Noether, to give some little contribution, and I shallnever forget the excitement of those days when the paper took shape.

The available documents indicate that a similar feeling of excitement was presentalso in the minds of the other actors in this play. Besides Hasse and Noether we haveto mention Artin and also Albert in this connection. Other names will appear in duecourse.

As toA.AdrianAlbert, he had an extended exchange of letters with Hasse, startingin 1931, on the Local-Global Principle for algebras. In the paper of Brauer–Hasse–Noether the authors explain that and how Albert had an independent share in theproof of the Main Theorem. Accordingly some people have suggested that perhapsit would be justified to include his name as an author, i.e., to talk about the “Albert–Brauer–Hasse–Noether Theorem”. But in this article we shall use the original nameof the theorem, i.e., without Albert, since this has become standard in the literature.In Section 1.8 we will describe the role of Albert in the proof of the Brauer–Hasse–Noether theorem, based on the relevant part of the correspondence of Albert withHasse.

Acknowledgement. Preliminary versions had been on my homepage for some time.I would like to express my thanks to all who cared to send me their comments eachof which I have carefully examined and taken into consideration. Moreover, I wishto thank Falko Lorenz and Keith Conrad for their careful reading, their correctionsand valuable comments. Last but not least I would like to express my gratitude toMrs. Nancy Albert, daughter of A.A.Albert, for letting me share her recollections ofher father. This was particularly helpful to me while preparing Section 1.8.

1.2 The Main Theorem: cyclic algebras

On December 29, 1931 Kurt Hensel, the mathematician who had discovered p-adic numbers, celebrated his 70th birthday. On this occasion a special volume ofCrelle’s Journal was dedicated to him since he was the chief editor of Crelle’s Journalat that time, and had been for almost 30 years. The dedication volume containsthe paper [BHN32], authored jointly by Richard Brauer, Helmut Hasse and EmmyNoether, with the title:

Beweis eines Hauptsatzes in der Theorie der Algebren


Proof of a Main Theorem in the theory of algebras

The paper starts with the following sentence:

Endlich ist es unseren vereinten Bemühungen gelungen, die Richtigkeitdes folgenden Satzes zu beweisen, der für die Strukturtheorie der Al-gebren über algebraischen Zahlkörpern sowie auch darüber hinaus vongrundlegender Bedeutung ist: …

At last our joint endeavours have finally been successful, to prove thefollowing theorem which is of fundamental importance for the structuretheory of algebras over number fields, and also beyond …

The theorem in question, which has become known as the Brauer–Hasse–NoetherTheorem, reads as follows:

Hauptsatz. Jede normale Divisionsalgebra über einem algebraischenZahlkörper ist zyklisch (oder, wiemanman auch sagt, vomDicksonschenTypus).

Main Theorem.2 Every central division algebra over a number field iscyclic (or, as it is also said, of Dickson type).

In this connection, all algebras are assumed to be finite dimensional over a field. AnalgebraA over a fieldK is called “central” ifK equals the center ofA. Actually, in theoriginal Brauer–Hasse–Noether paper [BHN32] the word “normal” was used insteadof “central”; this had gradually come into use at that time, following the terminology ofAmerican authors, see e.g., [Alb30].3 Today the more intuitive “central” is standard.

Cyclic algebras are defined as follows. Let LjK be a cyclic field extension, ofdegree n, and let � denote a generator of its Galois group G. Given any a in themultiplicative group K�, consider the K-algebra generated by L and some elementu with the defining relations:

un D a; xu D ux� .for x 2 L/:

This is a central simple algebra of dimension n2 overK and is denoted by .LjK; �; a/.The field L is a maximal commutative subalgebra of .LjK; �; a/. This construction

2Falko Lorenz [Lor05] has criticized the terminology “Main Theorem”. Indeed, what today is seen as a “MainTheorem” may in the future be looked at just as a useful lemma. So we should try to invent another name forthis theorem, perhaps “Cyclicity Theorem”. But for the purpose of the present article, let us keep the authors’terminology and refer to it as the “Main Theorem” (in capitals).

3It seems that in 1931 the terminology “normal” was not yet generally accepted. For, when Hasse had sentNoether the manuscript of their joint paper asking for her comments, she suggested that for “German readers”Hasse should explain the notion of “normal”. (Letter of November 12, 1931.) Hasse followed her suggestionand inserted an explanation.


of cyclic algebras had been given by Dickson; therefore they were also called “ofDickson type”.4

Thus the Main Theorem asserts that every central division algebraA over a numberfield K is isomorphic to .LjK; �; a/ for a suitable cyclic extension LjK with gen-erating automorphism � , and suitable a 2 K�; equivalently, A contains a maximalcommutative subfield L which is a cyclic field extension of K.

When Artin heard of the proof of the Main Theorem he wrote to Hasse:5

… Sie können sich gar nicht vorstellen, wie ich mich über den endlichgeglückten Beweis für die cyklischen Systeme gefreut habe. Das istder grösste Fortschritt in der Zahlentheorie der letzten Jahre. Meinenherzlichen Glückwunsch zu Ihrem Beweis. …

…You cannot imagine how ever so pleased I was about the proof, finallysuccessful, for the cyclic systems. This is the greatest advance in numbertheory of the last years. My heartfelt congratulations for your proof. …

Now, given the bare statement of the Main Theorem, Artin’s enthusiastic exclamationsounds somewhat exaggerated. At first glance the theorem appears as a rather specialresult. The description of central simple algebras may have been of importance, butwould it qualify for the “greatest advance in number theory in the last years”? It seemsthat Artin had in mind not only the Main Theorem itself, but also its proof, involvingthe so-called Local-Global Principle and its many consequences, in particular in classfield theory.

The authors themselves, in the first sentence of their joint paper, tell us that theysee the importance of the Main Theorem in the following two directions:

1. Structure of division algebras. The Main Theorem allows a complete classifica-tion of division algebras over a number field by means of what today are calledHasse invariants; thereby the structure of the Brauer group of an algebraicnumber field is determined. (This was elaborated in Hasse’s subsequent paper[Has33a] which was dedicated to Emmy Noether on the occasion of her 50thbirthday on March 23, 1932.) The splitting fields of a division algebra canbe explicitly described by their local behavior; this is important for the repre-sentation theory of groups. (This had been the main motivation for RichardBrauer in this project.)

2. Beyond the theory of algebras. The Main Theorem opens new vistas into oneof the most exciting areas of algebraic number theory at the time, namelythe understanding of class field theory – its foundation, its structure and its

4Dickson himself [Dic27] called these “algebras of type D”. Albert [Alb30] gives 1905 as the year whenDickson had discovered this construction. – Dickson did not yet use the notation .LjK; �; a/ which seems tohave been introduced by Hasse.

5This letter from Artin to Hasse is not dated but we have reason to believe that it was written around November11, 1931.


generalization – by means of the structure of algebras. (This had been suggestedfor some time by Emmy Noether. It was alsoArtin’s viewpoint when he praisedthe Main Theorem in the letter we have cited above.)

We will discuss these two viewpoints in more detail in the course of this article.

1.3 The paper: dedication to Hensel

But let us first have a brief look at the dates involved. The Hensel Festband of Crelle’sJournal carries the publication date of January 6, 1932. The first copy was finishedand presented to Hensel already on December 29, 1931, his birthday.6 The Brauer–Hasse–Noether paper carries the date of receipt of November 11, 1931. Thus the paperwas processed and printed within less than two months. This is a remarkably shorttime for processing and printing, including two times proofreading by the authors.It seems that the authors submitted their paper in the last minute, just in time tobe included into the Hensel dedication volume. Why did the authors not submit itearlier? After all, Hasse himself was one of the editors of Crelle’s Journal and so hewas informed well in advance about the plans for the Hensel dedication volume.

The answer to our question is that the authors did not find their result earlier. For wecan determine almost precisely the day when the proof of the Main Theorem had beencompleted. There is a postcard from Emmy Noether to Hasse dated November 10,1931 which starts with the following words:

Das ist schön! Und mir ganz unerwartet, so trivial der letzte Schluß ist;der ja auch bei Brauer steht (Jede Primzahl des Index geht im Exponentenauf.) …

This is beautiful! And completely unexpected to me, notwithstandingthat the last argument, due to Brauer, is quite trivial (Every prime numberdividing the index is also a divisor of the exponent.) …

This is a response to a postcard from Hasse telling her that he had found the last stepin the proof of the Main Theorem, by means of an argument which Hasse had learnedfrom Brauer. The theorem of Brauer which she cites in parentheses had been provedin [Bra29b]. Of course she does not mean that Brauer’s theorem is trivial, but that theapplication of Brauer’s theorem in the present situation seems trivial to her. Actually,we shall see in Section 1.4.2 that this theorem of Brauer is not really needed but onlyhis Sylow argument which he had used in [Bra29b].

Only two days earlier, on November 8, 1931, Noether had sent a long letter7 in6We know this because Hasse mentioned it in his laudation which he read to Hensel on the birthday reception.

See [Has32a].7The letter has four pages. This must be considered as “long” by the standard of Emmy Noether who often

scribbled her messages on postcards, using up every conceivable free space on the card.


which she congratulated Hasse for his recent proof that at least every abelian centralsimple algebra A is cyclic. Here, a central simple algebra AjK is called “abelian” ifit admits a splitting field which is an abelian field extension of K.

But Noether did not only congratulate. In addition, she showed Hasse how toobtain a simplification (which she called “trivialization”) of his proof, and at thesame time to generalize his result from “abelian” to “solvable” algebras by means ofan easy induction argument. Moreover, she gave some ideas how it may be possibleto approach the general, non-solvable case. These latter ideas were quite differentfrom the final solution which consisted in applying Brauer’s Sylow argument; thisexplains her surprise which she shows in her postcard of November 10.

In those times, postal mail went quite fast. Between Marburg (where Hasselived) and Göttingen (Noether’s place) ordinary mail was delivered the next day afterdispatch, sometimes even on the same day.8

Thus it appears that on November 9, Hasse had received Noether’s earlier letterof November 8. While studying her proofs for the solvable case he remembered anearlier letter of Brauer, where a Sylow argument was used to reduce the general caseto the case of a p-group which, after all, is solvable. Putting things together Hassesaw the solution. Brauer’s letter had been written some days earlier, on October 29.9

Immediately Hasse informed Emmy Noether about his finding, and so it waspossible that she received his message on November 10 and could send her replypostcard on the same day.

Accordingly we may conclude that November 9, 1931 is to be very likely thebirthday of the Brauer–Hasse–Noether Main Theorem, i.e., the day when the last stepin the proof had been found.

The same day Hasse informed Richard Brauer too. Just two days earlier, onNovember 7, Hasse had sent a long 10 page letter to Brauer, explaining to him inevery detail his ideas for attacking the problem. He used Brauer’s Sylow argumentbut then he said:

Leider muß ich bekennen, daß ich hier am Ende meines Könnens steheund alle meine Hoffnungen auf Ihr Können setze. Es handelt sich, wieSie sehen um ein Faktorensystem, das zu einem nicht-galoisschen Zer-fällungskörper gehört …

I have to admit that here I am at the end of my skills and I put all my hopeon yours. As you see, there is a factor system involved which belongsto a non-galois splitting field …

Since Brauer had introduced and investigated factor systems for non-galois splittingfields [Bra26], [Bra28], it appears quite natural that Hasse turned to him for the

8Mail was delivered two times a day: once in the morning and a second time in the afternoon.9Actually, Brauer in his letter did not have the Main Theorem in mind but the related question whether the

index of an algebra equals its exponent, over an algebraic number field as a base.


solution of the problem. But two days later Hasse could send a postcard with thefollowing text:

Lieber Herr Brauer ! Eben bekomme ich einen Brief von Emmy, der dieganze Frage erledigt, und zwar so, daß ein Eingehen auf die Strukturder Faktorensysteme gar nicht notwendig wird … man kann nämlichden Beweis durch schrittweise Reduktion in Primzahlschritten führen.Man muss nur den Abbau nicht, wie ich ungeschickt versuchte, beimKörper unten, bei der Gruppe oben beginnen, sondern umgekehrt … Ichhabe mich furchtbar gequält, und doch nicht den einfachen Gedankenvon Emmy gehabt.

Dear Mr. Brauer! Just now I receive a letter from Emmy which takescare of the whole question, and such that it will not be necessary toknow the structure of the factor systems … It is possible to get a proofby stepwise reduction to steps of prime degree. The only thing to do isnot, as I had clumsily tried, to start the reduction process with the fieldbelow, i.e., with the group on top, but to do it the other way … I had goneto many troubles but did not find the simple idea of Emmy.

And Hasse continued to describe Emmy’s idea, all on the same postcard.Brauer lived in Königsberg which was somewhat more distant from Marburg than

Göttingen; thus the postcard to him may have needed one day longer than that toNoether.10 In fact, Brauer’s reply to Hasse is dated November 11, one day later thanNoether’s reply. He wrote:

Herzlichen Dank für Ihren ausführlichen Brief und Ihre Karte, die icheben erhielt. Es ist sehr schön, daß das Zyklizitätsproblem jetzt erledigtist! Ich hatte Ihnen gerade heute schreiben wollen und Ihnen genau dieMethode der Emmy mitteilen wollen; allerdings muß ich offen sagen,daß ich fürchtete, einen dummen Fehler dabei zu machen, weil mirdie Sache zu einfach vorkam. Ich hatte Sie gerade deswegen anfragenwollen, was ja nun überflüssig geworden ist. Es war mir übrigens vonvornherein klar, daß durch Ihre Reduktion die wesentliche Arbeit gelei-stet war.

Many thanks for your detailed letter, and for your postcard which I justreceived. It is very nice that the problem of cyclicity is now solved!Just today I had meant to write you and to inform you in detail aboutEmmy’s method; but I have to admit that I feared to make a silly mistakebecause I had the feeling that the thing was too simple. I just wantedto ask you about it, but now this is unnecessary. By the way, right from

10From Marburg to Göttingen there are about 140 km, whereas from Marburg to Königsberg we have countedabout 975 km.


the beginning it was clear to me that with your reduction, the essentialwork had been done already.

This shows that Brauer was directly involved in finding the proof. When hementions “Emmy’s method” he refers to Hasse’s postcard where Hasse had explainedthe reduction step sent to him by Emmy Noether. But in fact, Brauer had found thesame method independently and he too had realized that, if combined with his Sylowargument, this method would give the solution. At the same time we see his modesty,which made him claim that Hasse had done the essential work already. Two dayslater when Hasse had sent him the completed manuscript, he wrote:

Heute früh erhielt ich Ihre Arbeit; ich bin ganz überrascht, daß meinedoch wirklich nur geringfügige Bemerkung Sie veranlaßt hat, diesebesonders schöne Arbeit mit unter meinem Namen zu publizieren.

Today in the morning I received your paper; I am quite surprised that myreally small remark has caused you to publish this particularly beautifulpaper jointly under my name.

Well, Brauer’s contribution was not confined to a “small remark”. On the contrary,Hasse’s arguments relied heavily and substantially on Brauer’s general results aboutdivision algebras and their splitting fields.

We have seen that the “birthday” of the Main Theorem had been November 9, butwe have also seen that the manuscript was received by the editors on November 11.We conclude that Hasse had completed the manuscript in at most two days. Actually, itmust have been within one day because on November 11 already, Emmy Noether hadreceived from him the completed manuscript and wrote another letter to Hasse withher comments. This haste is explained by the fact that the deadline for contributionsto the Hensel dedication volume had passed long ago (it was September 1, 1931) andHensel’s birthday was approaching at the end of the year already, when the volume hadto be presented to him. And Hasse was eager to put this paper, which he consideredimportant, into this dedication volume. Kurt Hensel had been his respected academicteacher and now was his paternal friend (“väterlicher Freund”). In the introductionof the Brauer–Hasse–Noether paper we read:

Es ist uns eine besondere Freude, dieses Ergebnis, als einen im wesent-lichen derp-adischen Methode zu dankenden Erfolg, Herrn Kurt Hensel,dem Begründer dieser Methode, zu seinem 70. Geburtstag vorzulegen.

It gives us particular pleasure to be able to dedicate this result, beingessentially due to the p-adic method, to the founder of this method,Mr. Kurt Hensel, on the occasion of his 70th birthday.

Emmy Noether commented on this dedication text in her letter of November 12 toHasse as follows:


Mit der Verbeugung vor Hensel bin ich selbstverständlich einverstanden.Meine Methoden sind Arbeits- und Auffassungsmethoden, und daheranonym überall eingedrungen.

Of course I agree with the bow to Hensel. My methods are working-and conceptual methods and therefore have anonymously penetratedeverywhere.

The second sentence in this comment has become famous in the Noether literature. Itputs into evidence that she was very sure about the power and success of “her methods”which she describes quite to the point. But why did she write this sentence just here,while discussing the dedication text for Hensel ? The answer which suggests itself isthat, on the one hand, Noether wishes to express to Hasse that, after all, “her methods”(as distinguished from Hensel’s p-adic methods) were equally responsible for theirsuccess. On the other hand she does not care whether this is publicly acknowledgedor not.

In the present context “her methods” means two things: First, she insists that theclassical representation theory be done within the framework of the abstract theoryof algebras (or hypercomplex systems in her terminology), instead of matrix groupsand semi-groups as Schur had started it. Second, she strongly proposes that thenon-commutative theory of algebras should be used for a better understanding ofcommutative algebraic number theory, in particular class field theory.

Perhaps we may add a third aspect of “her methods”: the power to transmit herideas and concepts to the people around her. In this way she had decisively influencedRichard Brauer’s and Helmut Hasse’s way of thinking: Brauer investigated divisionalgebras and Hasse did non-commutative arithmetic.

The great hurry in which the Brauer–Hasse–Noether paper had to be written mayalso account for the somewhat unconventional presentation. For, Hasse says in afootnote that the material is presented

… in der Reihenfolge ihrer Entstehung, die der systematischen Reihen-folge entgegengesetzt ist …

… in the order of the discovery, which is the reverse of the systematicorder …

This footnote was inserted on the insistence of Noether. For, in still another letterwritten 3 days later, on November 14, 1931, she had expressed her dislike of thepresentation as given by Hasse. She wrote that in this presentation the proof is difficultto understand, and that she would have insisted on a more systematic arrangementexcept that the time was too short. Therefore Hasse should at least insert a footnote tothe effect as mentioned above. And Hasse did so. He wished the paper to be includedinto the Hensel volume, hence there was no time to rewrite the manuscript.

Three months later Hasse seized an opportunity to become reconciled with EmmyNoether by dedicating a new paper [Has33a] to her, on the occasion of her 50th


birthday on March 23, 1932. There he deals with the same subject but written moresystematically. Those three months had seen a rapid development of the subject; inparticular Hasse was now able to give a proof of Artin’s Reciprocity Law of classfield theory which was based almost entirely on the theory of the Brauer group over anumber field. Thereby he could fulfill a desideratum of Emmy Noether who alreadyone year earlier had asked him to give a hypercomplex foundation of the reciprocitylaw. In the preface to that paper Hasse “bows” to Emmy Noether as an invaluablesource of inspiration.

Section II.6 of that paper [Has33a] contains a new presentation of the Main Theo-rem. Hasse starts this section by admitting that in the earlier joint paper [BHN32]the proof had been presented in a somewhat awkward manner, according to the orderof its discovery. Now, he says, he will give the proof (which is the same proof afterall) in a more systematic way. Clearly, this is to be viewed as a response to Noether’scriticism in her letter of November 14.

By the way, three days after her birthday Emmy Noether replied to this present:“I was terribly delighted!…” (Ich habe mich schrecklich gefreut! …). There followtwo pages of detailed comments to Hasse’s paper, showing that she had studied italready in detail.

1.4 The Local-Global Principle

LetK be an algebraic number field of finite degree. For every prime p ofK, finite orinfinite, let Kp denote the p-adic completion of K. For an algebra A over K we putAp D A ˝K Kp, the completion (also called localization) of A at p. An importantstep in the proof of the Main Theorem is the celebrated

Local-Global Principle for algebras. Let K be a number field andAjK be a central simple algebra. If ApjKp splits for every p then AjKsplits.

Here, “splitting” of AjK means that A is a full matrix algebra over K. Note thatthe Local-Global Principle is formulated for simple algebras, not only for divisionalgebras as the Main Theorem had been. Quite generally, it is more convenient to workwith simple algebras and, accordingly, formulate and prove the Main Theorem forsimple algebras instead of division algebras only. By Wedderburn’s theorem, everysimple algebra AjK is isomorphic to a full matrix ring over a division algebra DjK,andD is uniquely determined byA up to isomorphisms. Two central simple algebrasover K are called “similar” if their corresponding division algebras are isomorphic.

We shall discuss in Section 1.5 how the Local-Global Principle was used in theproof of the Main Theorem. In the present section we review the long way whichfinally led to the conception and the proof of the Local-Global Principle.


1.4.1 The Norm Theorem

First, consider a cyclic algebraA D .LjK; �; a/ as explained in Section 1.2. Such analgebra splits if and only if a is a norm from the cyclic extension LjK. Accordingly,the Local-Global Principle for cyclic algebras can be reformulated as follows:

Hilbert–Furtwängler–Hasse Norm Theorem. Let LjK be a cyclicextension of number fields, and let 0 ¤ a 2 K. If a is a norm in thecompletion LpjKp for every p then a is a norm in LjK.11

This theorem does not refer to algebras, it concerns algebraic number fields only.Now, in the case when the degree n of LjK is a prime number, the Norm Theoremwas known for a long time already, in the context of the reciprocity law of classfield theory. It had been included in Hasse’s class field report, Part II [Has30a]where Hasse mentioned that it had first been proved by Furtwängler in [Fur02] andsubsequent papers. For quadratic fields .n D 2/ the Norm Theorem had been givenby Hilbert in his Zahlbericht [Hil97]. In March 1931 Hasse succeeded to generalizethis statement to arbitrary cyclic extensions LjK of number fields, not necessarily ofprime degree; see Section 1.7. He published this in [Has31a], April 1931.

Now, the Main Theorem tells us that every central simple algebra over a numberfield is cyclic, so we could conclude that the Local-Global Principle holds generally,for every central simple algebra over a number field. However, in order to provethe Main Theorem, Hasse needed first to prove the Local-Global Principle generally,regardless of whether the given algebra is already known to be cyclic or not. Hencethere arose quite naturally the problem how to reduce the general case of the Local-Global Principle to the case when the algebra is cyclic.

1.4.2 The reductions

In the Brauer–Hasse–Noether paper [BHN32] this reduction is done in two steps:

(2) Reduction to the case when A has a solvable splitting field.(3) Further reduction (by induction) to the case when A has a cyclic

splitting field.

Here and in the following we use the same enumeration of these “reductions” whichis used in the Brauer–Hasse–Noether paper. There is another reduction, called “re-duction (1)”, which reduces the Main Theorem to the Local-Global Principle. Thatwe will discuss in Section 1.5 and the following sections, as the contribution of Hasse.

11In accordance with the definition of Ap one would define Lp D L ˝K Kp. In general this is not a fieldbut the direct sum of fields LP where P ranges over the primes of L dividing p. If LjK is a Galois extension(in particular if it is cyclic) then all these fields LP are isomorphic over Kp, and a is a norm from Lp if andonly if it is a norm from LP for some and hence all P. Usually, one chooses one prime Pjp and writes Lp forLP (thus forgetting the former, systematic notation for Lp). Let us do this here too.


The reduction (2) is due to Brauer who, in his letter to Hasse of October 29, 1931,had provided a Sylow argument for this purpose. Reduction (3) had been providedby Noether in her letter to Hasse of November 8, 1931.

Brauer had developed the theory of division algebras and matrix algebras in a seriesof several papers in the foregoing years, starting from his 1927 Habilitationsschriftat the University of Königsberg [Bra28]. His main interest was in the theory ofgroup representations, following the ideas of his academic teacher I. Schur. It wasEmmy Noether who gradually had convinced him that the representation theory ofgroups could and should be profitably discussed within the framework of algebras.In Brauer’s papers, in particular in [Bra29b], we find the following theorems. Brauerhad reported on these theorems in September 1928 at the annual meeting of the DMV(Deutsche Mathematiker Vereinigung) in Hamburg; see [Bra29a]. Although in thatreport no proofs are given, we can recommend consulting it since Brauer’s theoremsare very clearly stated there.

Brauer’s theorems

(i) The similarity classes of central simple algebras over a fieldK forma group with multiplication well-defined by the tensor product A˝K B

of two algebras.12

Today this group is called the “Brauer group” ofK and denoted by Br.K/. The name“Brauer group” was given by Hasse in [Has33a]. The split algebras belong to theneutral element of the Brauer group.

(ii) Every central simple algebra A over K has finite order in Br.K/.

This order is called the “exponent” ofA. This terminology had been chosen by Brauerbecause, he said, in the context of the theory of algebras the word “order” is used foranother concept.13

(iii) The exponent of A divides the index m of A.

The index m of A is defined as follows: Let D be the division algebra similar toA. The dimension of D over its center is a square m2, and this m is the index bydefinition.

(iv) Every prime number dividing the index ofA also divides its exponent.

Brauer had used these theorems (i)–(iv) in order to show:

12At the time of Brauer–Hasse–Noether, the tensor product was called “direct product” and denoted by A�B . –Brauer considered only perfect base fields K; it was Emmy Noether who in [Noe29] was able to wave thehypothesis that K is perfect.

13I am indebted to Falko Lorenz who pointed out to me that this theorem (ii) is contained in Schur’s paper[Sch19] already, as well as theorem (iii) if m is interpreted suitably. See [Lor98].


(v) Every division algebraA of indexm can be decomposed as the tensorproduct of division algebras Ai of prime power index p�i

i , according tothe prime power decomposition m D Q

i p�i

i of the index.

In Hasse’s first draft of the joint manuscript which he had sent to Emmy Noether,these theorems were used. Although we do not know this first draft, we can concludethis from the following: First, in Noether’s reply postcard of November 10 (which wehave cited in Section 1.3) Brauer’s theorem (iv) is mentioned. Secondly, in a letter ofHasse to Brauer dated November 11, Hasse reports that Noether had finally thrownout the reduction (v) to prime power index, because that was superfluous. And soHasse continues:

… Daher fand ich auch eigentlich nicht genug Gelegenheit Ihre Arbeitaus Math. Zeitschr. zu zitieren. Es wird fast nichts daraus gebraucht,außer den einfachsten schon vorher feststehenden Tatsachen über Zer-fällungskörper.

… For that reason I did not find a suitable occasion to cite your paper ofMathematische Zeitschrift. Almost nothing from there is needed, exceptthe most simple facts on splitting fields.

Here Hasse refers to Brauer’s paper [Bra29b]. These “most simple facts” whichare used in the final proof are the following:

(vi) The degree of every splitting field of A over K is divisible by theindex m of A, and there exist splitting fields of degree m.

Using this, the reduction steps (2) and (3) are quite easy if combined with thefunctorial properties of the Brauer group. Let us briefly present the arguments. Ourpresentation is the same as Noether had proposed it in her letter to Hasse of Novem-ber 10, 1931, and which Hasse then used in his Noether dedication paper [Has33a].

If K � L then we use the notation AL D A ˝K L. If we regard A and AL intheir respective Brauer groups Br.K/ and Br.L/ then the map A 7! AL defines acanonical group homomorphism Br.K/ ! Br.L/.

LetK be a number field andAjK a central simple algebra which splits everywherelocally. The claim is that A splits. Suppose A does not split and let m > 1 be theindex of A. Let p be a prime number dividing m. Consider a Galois splitting fieldLjK of A, so that AL splits; then p divides ŒL W K�. Let G be the Galois group ofLjK. Consider a corresponding Sylow p-group ofG and letL0 � L denote its fixedfield. Since the Sylow p-group is solvable there exists a chain of fields

L0 � L1 � � � � � Ls�1 � Ls D L

such that each Li jLi�1 is cyclic of degree p .1 � i � s/. Since A splits everywherelocally, so does every ALi

. Now, ALs�1has Ls D L as a cyclic splitting field.


Hence the Norm Theorem implies that ALs�1splits. Therefore ALs�2

has Ls�1 as acyclic splitting field, hence again, ALs�2

splits. And so on by induction. Finally weconclude thatAL0

splits. ThusA admits the splitting fieldL0 whose degree ŒL0 W K�is relatively prime to p. But ŒL0 W K� is divisible by the index m which contains pas a prime divisor. Contradiction.

Since eachLi jLi�1 is of degree p, it is evident that Hasse’s Norm Theorem has tobe used only in the case of cyclic fields of prime degree p, i.e., the original Hilbert–Furtwängler Theorem is sufficient. Hasse’s generalization to arbitrary cyclic fieldsis not needed and is a consequence of Noether’s induction argument. This had beenimmediately observed by Noether (letter of November 8, 1931), and she had askedHasse to mention it in their joint paper (which he did). At that time this observationindeed could be considered a simplification. But half a year later, in [Has33a], Hasseremarked that this would not make a difference any more because in the meantimenew proofs of the Norm Theorem had been found by Chevalley and Herbrand, andthose proofs work equally well for arbitrary cyclic extensions (using the so-calledHerbrand’s Lemma) regardless of whether the degree is prime or not.14

From today’s viewpoint the above proof of the Local-Global Principle looks rathertrivial, once the Hilbert–Furtwängler Norm Theorem is accepted. In particular if thearguments are given in the language of cohomology, as it is usually done nowadays,we see that only the very basic properties of the cohomological restriction map areused. This seems to justify Brauer’s words, cited above, that “right from the beginningit was clear to me [Brauer] that with your Hasse’s reduction, the essential work hadbeen done already”. But these words are valid only if, firstly, Brauer’s fundamentaltheorems are accepted and, secondly, there had already developed a certain routinefor using those theorems for particular problems. While the first was certainly thecase within the circle around Brauer, the second was not yet. Otherwise, the simpleproof above could well have been given much earlier.

We should not underestimate the conceptual difficulty which people had at thattime working with algebras and their splitting fields, and the notions of index andexponent of algebras. There was no established routine to work with the functorialproperties of Brauer groups. Based on the cited work of Brauer and, in parallel, onthe monumental work of Emmy Noether [Noe29] such routine came gradually intobeing.

1.4.3 Factor systems

The idea for a proof like the above, required in the first place some insight into therelevant structures, in particular the interpretation of the Norm Theorem as a splittingtheorem for cyclic algebras. Only under this aspect it makes sense to generalize itfrom the cyclic to the general case. In fact, Hasse originally did not do this step. In his

14The Herbrand–Chevalley proof was included in Hasse’s Marburg lectures 1932 on class field theory. See[Has33b], Satz (113).


class field report Part II [Has30a] he had conjectured that the Norm Theorem holdsfor arbitrary abelian extension of number fields. But in [Has31a] he had to admit thatfor non-cyclic extensions the Norm Theorem fails to hold.

It was Emmy Noether who then suggested to Hasse that the generalization ofthe Norm Theorem would require considering algebras instead of norms, the latterrepresenting split cyclic algebras. This is evidenced by the following excerpt fromher letter of November 12, 1931. In that letter she wished to have some furtherchanges in the manuscript of the joint Brauer–Hasse–Noether paper, for which Hassehad composed the draft. She wrote:

Ebenso möchte ich auf S. 4, im 4.-letzten Absatz, mitgenannt sein, oderetwa das H. Hasse durch „wir“ ersetzt haben. Daß nämlich die Fassungmit den Faktorensystemen die richtige Verallgemeinerung ist, habe ichIhnen schon auf dem Hanstein-Spaziergang im Frühling gesagt, als Siemir die Widerlegung der Norm-Vermutung im Abelschen Fall erzählten.Sie haben es damals wahrscheinlich noch nicht ganz aufgefaßt; und essich später selbst wieder überlegt. Genau genommen habe ich es Ihnenschon in Nidden gesagt.

Similarly, I would like to be mentioned too on page 4, in the 4th paragraphfrom below, or maybe the “H. Hasse” should be replaced by “we”15. For,I have mentioned to you already in the spring on our Hanstein-walk16

that the version with factor systems is the correct generalization, afteryou had told me the refutation of the norm conjecture in the abeliancase. Perhaps you had not yet fully grasped it at the time, and later youhave come to the same conclusion by yourself. Strictly speaking I hadmentioned this to you already in Nidden.17

We observe that Noether talks about factor systems and not about algebras. Factorsystems are used to construct algebras. Given any finite separable field extensionLjKlet Br.LjK/ denote the kernel of the map Br.K/ ! Br.L/, consisting of those centralsimple algebras over K (modulo similarity) which are split by L. Brauer had shownthat Br.LjK/ is isomorphic to the group of what he called factor systems (moduloequivalence). A factor system consists of certain elements in the Galois closure ofLjK, and it can be used to construct a central simple algebra AjK such that theelements of the factor system appear as factors in the defining relations of a suitable

15In the printed version, this is the last paragraph of section 4. There indeed we find the word “we” as Noetherrequested.

16Hanstein is a hillside near Göttingen. It appears that in the spring of 1931, on one of the many visits ofHasse to Göttingen, they had made a joint excursion to the Hanstein.

17Nidden at that time was a small fisherman’s village in East Prussia, located on a peninsula (KurischeNehrung) in the Baltic sea and famed for its extended white sand dunes. In September 1930, Hasse and Noetherboth attended the annual meeting of the DMV at Königsberg in East Prussia, and after the meeting they visitedNidden.


basis of the algebra. Brauer’s invention of factor systems was essential for the proofof his theorems.

It is true that the appearance of factor systems had been observed earlier already bySchur and also by Dickson. But it was Brauer who defined and used them systemati-cally to construct algebras, thereby writing down explicitly the so-called associativityrelations.

We will not give here the explicit definition of factor systems in the sense ofBrauer. For, today one mostly uses in this context the simplified form which Noetherhas given to Brauer’s factor systems. Noether considered Galois splitting fields LjKonly. Let G D G.LjK/ denote its Galois group. Consider theK-algebra A which isgenerated by L and by elements u� (� 2 G) with the defining relations

u�u� D u��a�; � ; xu� D u�x� .for x 2 L/

where �; � 2 G and a�; � 2 K�. It is required that the factors a�; � satisfy thefollowing relations which are called associativity relations:

a�; �% � a�; % D a��; % � a%�; � :

Sometimes they are also called Noether equations. The algebra A thus defined is acentral simple algebra over K which has L as a maximal commutative subfield. Ais called the “crossed product” of L with its Galois group G, and with factor systema D .a�; � /. 18 Notation: A D .LjK; a/. Every central simple algebra over K whichadmits L as a splitting field can be represented, up to similarity, as a crossed productin this sense. IfG is cyclic then (by appropriate choice of the u� ) we obtain the cyclicalgebras in this way.

This theory of factor systems was developed by Emmy Noether in her Göttingenlecture 1929/30. But Noether herself never published her theory. Deuring took notesof that lecture, and these were distributed among interested people; Brauer as wellas Hasse had obtained a copy of those notes. (The Deuring notes are now includedin Noether’s Collected Papers.) The first publication of Noether’s theory of crossedproducts was given, with Noether’s permission, in Hasse’s American paper [Has32c]where a whole chapter is devoted to it. The theory was also included in the bookAlgebren by Deuring [Deu35].

A factor system a�; � is said to split if there exist elements c� 2 L� such that

a�; � D c��c�

c��

:

18The German terminology is verschränktes Produkt. The English term crossed product had been used byHasse in his American paper [Has32c]. When Noether read this she wrote to Hasse: “Are the ’crossed products’your English invention? This word is good.” We do not know whether Hasse himself invented this terminology,or perhaps it was H. T. Engstrom, the American mathematician who helped Hasse to translate his manuscriptfrom German into English. In any case, in the English language the terminology “crossed product” has been inuse since then.


The split factor systems are those whose crossed product algebra .LjK; a/ splits. To-day we view the group of factor systems modulo split ones as the second cohomologygroup of the Galois groupG ofLjK in the multiplicativeG-moduleL�. The notationis H 2.G;L�/ or better H 2.LjK/.

Thus the Brauer–Noether theory of crossed products yields an isomorphism

Br.LjK/ � H 2.LjK/which has turned out to be basic for Brauer’s theory.

In mathematics we often observe that a particular object can be looked at from dif-ferent points of view. A change of viewpoint may sometimes generate new analogies,thereby we may see that certain methods had been successfully applied in similarlylooking situations and we try to use those methods, suitably modified, to deal also withthe problem at hand. This indeed can lead the way to new discoveries. But sometimesit can also hamper the way because the chosen analogies create difficulties which areinessential to the original problem.

We can observe such a situation in Hasse’s first attempts to deal with the Local-Global Principle for algebras. Instead of dealing with algebras directly he considered,following Noether’s suggestion, factor systems. Given a factor system in H 2.LjK/which splits locally everywhere, he tried to transform it in such a way that its globalsplitting is evident. This then reduces to the solution of certain diophantine equationsin L under the hypothesis that those equations can be solved locally everywhere.Now it is well known, and it was of course known to Hasse that the local solutionof diophantine equations does not in general imply their global solution. But severalyears earlier Hasse had already proved one instance of a Local-Global Principle forcertain diophantine equations, namely quadratic equations. In case of the rationalfield Q as base field this had been the subject of Hasse’s dissertation (Ph.D. thesis)in 1921, and in subsequent papers [Has24b], [Has24a] he solved the same problemfor an arbitrary number field K as base field.19

Accordingly, Hasse tried first to invoke the analogy to the theory of quadraticforms in order to approach the Local-Global Principle for algebras. However, itturned out that this created difficulties which only later were seen not to be inherentto the problem.

We are able to follow Hasse’s ideas for these first attempts (which later werediscarded as unnecessary) since Hasse had written to some of his friends explaining

19With this result Hasse had solved, at least partially, one of the famous Hilbert problems. The 13th Hilbertproblem calls for solving a given quadratic equation with algebraic numerical coefficients in any number ofvariables by integral or fractional numbers belonging to the algebraic realm of rationality determined by thecoefficients. Hilbert’s wording admits two interpretations. One of them is to regard the phrase “integer orfractional numbers” as denoting arbitrary numbers of the number field in question. In this interpretation Hassecould be said to have solved the problem completely. The other interpretation is that Hilbert actually meant twodifferent problems: The first is to solve the quadratic equation in integers of the field, and the second is to admitsolutions with arbitrary numbers of the field. In this interpretation, which would generalize Minkowski’s workfrom the rationals to arbitrary number fields, Hasse would have solved only the second of the two problems. Thefirst problem (solution in integers) has been studied by Siegel and others.


these ideas, obviously in the hope that someone would be able to supply the finalclue. One of those letters, the one to Brauer dated July 27, 1931, is preserved. Hassewrites:

Ich möchte Ihnen gerne schreiben, wie die Sachlage nun mit der einzi-gen noch offenen Frage nach der Zyklizität aller normalen einfachenAlgebren steht. Ich glaube nämlich, daß diese Frage jetzt angriffsreifist, und möchte Ihnen die mir vorschwebende Angriffslinie vorlegen.

I would like to write to you about the only question which is still open, thequestion whether all central20 simple algebras are cyclic. For I believethat this question is now ripe and I would like to present to you the lineof attack which I have in mind.

(In this connection Hasse means algebras over an algebraic number field as base field,although he does not explicitly mention this.)

Hasse continues that, following his “line of attack”, he is trying to use his Local-Global Principle for quadratic forms. Let wi be a basis of the given central simplealgebra A over K. The trace matrix tr.wiwk/ defines a quadratic form. If A splitslocally everywhere then for every prime p there exists a basis transformation whichtransforms the given basis into a system of matrix units, and this defines a certaintransformation of the quadratic form. The Local-Global Principle for quadratic formsthen yields a certain basis transformation over the global fieldK. Hasse asks whetherit is possible to deduce the splitting ofA from the special structure of this transformedtrace form. In other words, one has to construct from it a system of matrix units overK. But Hasse does not yet know how to do this, not even whether it is possible at all.He writes to Brauer:

Ich möchte diese Sache zur Überprüfung nach diesem Gesichtspunkt inIhre kundigen Hände legen.

I would like to put this problem into your hands for examination fromthis viewpoint.

Several days later, on August 3, 1931 Brauer replied that at present he is not able tosay anything about Hasse’s problem, and that first he has to study it in detail. Butrelying on Hasse’s own creative power he adds:

Ich hoffe wenigstens so weit zu sein, daß ich alles verstehen kann, wennSie selbst die Lücke ausgefüllt haben werden.

I hope to be able to understand all these things at the time when you willhave filled the gap yourself.

20Hasse writes “normal” instead of “central”. For the convenience of the reader we will replace “normal” inthis context by the modern “central”, here and also in other citations which follow.


Hasse had sent similar letters to Artin and Noether. These letters are not preserved butwe know the respective answers. Artin, returning from a vacation in the mountainswrote on August 24, 1931:

Inzwischen haben Sie sicher den Satz über Schiefkörper bewiesen. Ichbin schon sehr gespannt darauf.

… Meanwhile you will certainly have proved the theorem on divisionalgebras. I am looking forward to it.

Noether wrote on the same day:

Natürlich kann ich Ihre Frage auch nicht beantworten – ich glaube manmuß so etwas liegen lassen bis man von anderer Seite selbst daraufstößt …

Naturally, I too cannot answer your question – I believe one shouldleave such things alone until one meets them again from another pointof view …

But she adds some remarks about the work of Levitzky (her Ph.D. student) whoprovided some methods to construct bases of split algebras.

These answers do not sound as if they had been very helpful to Hasse. But he didnot give up so easily. After a while he managed to prove the Local-Global Principlefor those algebras A which admit an abelian splitting field LjK. We do not knowthis proof but from Noether’s reaction we can infer that indeed Hasse had explicitlyconstructed, by induction, a split factor system for the algebra. We have alreadymentioned in Section 1.3 (p. 7) Noether’s reaction to Hasse’s letter; the Noether letterwas dated November 8, 1931 and gave a simplification and generalization of Hasse’sresult to algebras which admit a solvable splitting field, not necessary abelian.

From then on things began to develop rapidly as we have explained in Section 1.3,and one day later the proof of the Local-Global Principle was complete.

As a side remark we mention that Hasse in his letter to Brauer of November 16,1931 states that when Noether’s postcard arrived on November 9 he had “essentiallybeen through” with his complicated proof. But, as we have seen, he immediatelythrew away his complicated proof in favor of Noether’s “trivialization”.

1.5 From the LGP to the Main Theorem

Sometimes the Local-Global Principle is considered the most important result of theBrauer–Hasse–Noether paper while the Main Theorem is rated as just one of themany consequences of it. But the authors themselves present the Main Theorem astheir key result. We now discuss the step from the Local-Global Principle to the


Main Theorem. This is the “reduction (1)” in the count of the Brauer–Hasse–Noetherpaper, and it is due to Hasse.

1.5.1 The Splitting Criterion

Let A be a central simple algebra over a number field K. Then ŒA W K� is a square;let ŒA W K� D n2 with n 2 N. It is known that n is a multiple of the indexm of A. Inorder to show that A is cyclic one has to construct a cyclic splitting field LjK of Aof degree ŒL W K� D n. To this end one needs a criterion for a finite extension fieldL of K to be a splitting field of A.

According to the Local-Global Principle the problem can be shifted to the localcompletions, namely:

A is split by L if and only if each Ap is split by LP for Pjp.

In the local case, there is a simple criterion for splitting fields:

Local Splitting Criterion. Ap is split by LP if and only if the degreeŒLP W Kp� is divisible by the index mp of Ap.

Thus the Local-Global Principle yields:

Global Splitting Criterion. A is split byL if and only if for each primep ofK and each P dividing p the local degree ŒLP W Kp� is divisible bythe local index mp of Ap.

If LjK is a Galois extension then for all primes P dividing p the completions LP

coincide; they may be denoted by Lp according to the notation explained in foot-note 11.

The local criterion was essentially contained in Hasse’s seminal Annalen paper[Has31d] on the structure of division algebras over local fields. But the criterionwas not explicitly stated there. Therefore Hasse in their joint paper [BHN32] gave adetailed proof of the criterion, based on the main results of [Has31d]. But again itwas not explicitly stated; instead, the statement and proof was embedded in the proofof the global criterion which was “Satz 3” in the joint paper.

So the local criterion, although it is one of the basic foundations on which theMain Theorem rests, remained somewhat hidden in the Brauer–Hasse–Noether paper– another sign that the preparation of the manuscript was done in great haste. It wasso well hidden that even five months later Emmy Noether was not aware that its proofwas contained in the paper of which she was a co-author after all. In her letter ofApril 27, 1932 she wrote, referring to a recent paper of Köthe:21

21Gottfried Köthe (1905–1989) was a young post-doc who in 1928/1929 came to Göttingen to study mainlywith Emmy Noether and van der Waerden. Later he switched to functional analysis under the influence ofToeplitz.


… Tatsächlich zeigt Köthe mit seinem Invariantensatz ja direkt, daß imp-adischen die Gradbedingung auch hinreichend ist für Zerfällungskör-per …

… In fact, Köthe with his theorem on invariants shows directly that in thep-adic case the degree condition is also sufficient for splitting fields …

With “invariants” are meant what today are called the “Hasse invariants” of centralsimple algebras over a local fieldKp (see Section 1.6.1, p. 35 below). Köthe’s theoremin [Köt33] describes the effect of a base field extension to these invariants. If the basefield Kp is extended to a finite extension Lp then, according to Köthe’s theorem, theHasse invariant of the extended algebra ALp is obtained from the Hasse invariant ofAp by multiplication with the field degree ŒLp W Kp�. This implies the local splittingcriterion.

One week later Noether admitted that she had overlooked Hasse’s proof in thejoint paper [BHN32]. Obviously responding to a reproach of Hasse she wrote:

… Als ich nun Köthe in die Hand bekam, fiel mir als erstes auf, daß jetztdiese alte Frage ja beantwortet ist. Bei Ihnen hatte ich drüber weggelesen;oder was wahrscheinlicher ist, ich dachte an meinen alten Beweis undhabe bei Ihnen im wesentlichen überflogen.

… When I got Köthe’s paper it occurred to me that now this old questionwas settled. In yours I had overlooked it; or, what is more likely, Ithought about my old proof and had only skimmed through yours.22

Now let us return to the Global Splitting Criterion. Its degree conditions arenon-trivial only for the primes p for which the local index mp > 1. For a givencentral simple algebra there are only finitely many such primes. This is by no meanstrivial; it had been proved by Hasse in [Has31d] where he showed that the reduceddiscriminant (“Grundideal”) of a maximal order of A contains p to the exponentmp � 1. We conclude that the existence of a cyclic splitting field LjK of degree nfor A is equivalent to the following general

Existence Theorem. Let K be an algebraic number field and S afinite set of primes of K. For each p 2 S let there be given a numbermp 2 N.23 Moreover, let n 2 N be a common multiple of the mp’s.Then there exists a cyclic field extension LjK of degree n such that foreach p 2 S the local degree ŒLp W Kp� is a multiple of mp.

22Since Noether had wished to inform Hasse about Köthe’s results it seems that she did not know (or notremember) that Köthe’s paper [Köt33] was written largely under the influence and the guidance of Hasse. This isexpressed by Köthe in a footnote to his paper which appeared in the Mathematische Annalen right after Hasse’s[Has33a].

23For infinite primes the usual restrictions should be observed: If p is real then mp D 1 or 2; if p is complexthen mp D 1. This guarantees that in any case mp is the index of some central simple algebra over Kp.


This then settles the Main Theorem.The Existence Theorem as such does not refer to algebras. It belongs to algebraic

number theory. We shall discuss the theorem and its history in the next sections.

1.5.2 An unproven theorem

A proof of the Existence Theorem had been outlined in a letter of Hasse to Albertwritten in April 1931. This is reported in the paper [AH32]. But the proof is notgiven and not even outlined in [AH32]. Hasse did not publish a proof of his existencetheorem, not in the joint paper [BHN32] and not elsewhere. Why not? After all, theexistence theorem is an indispensable link in the chain of arguments leading to theproof of the Main Theorem. Without it, the proof of the Main Theorem would beincomplete. Now, in a footnote in [AH32] we read:

The existence theorem is a generalization of those in Hasse’s papers[Has26c], [Has26b] and a complete proof will be published elsewhere.

This remark gives us a clue why Hasse may have hesitated to publish his proofprematurely. He regarded his existence theorem as an integral part of number theoryand was looking for the most general such theorem, independently of its applicationto the proof of the Main Theorem. We shall see that Grunwald, a Ph.D. student ofHasse, provided such a very general theorem. This then leads to the Grunwald–Wangstory.

The story begins with a reference which Hasse had inserted in the Brauer–Hasse–Noether paper [BHN32] for a possible proof of the Existence Theorem. This referencereads: “[vgl.d.Anm.zu H,17Bb]”. This somewhat cryptical reference can be decodedas: “compare the footnote in the paper H, section 17, Proof of (17.5) part B, subsec-tion (b).” The code “H” refers to Hasse’sAmerican paper [Has32c] on cyclic algebras.That paper had not yet appeared at the time when he wrote down the manuscript forthe Brauer–Hasse–Noether paper, hence he could not give a page number. We havechecked that the page number is 205. But there, in the said footnote of [Has32c] itis merely stated: “The existence of such a field will be proved in another place.”24

This does not sound very helpful to the reader.Let us check the next paper of Hasse [Has33a]. This is the one which he had

dedicated to Emmy Noether and in which, among other topics, he repeats the proofof the Main Theorem more systematically. There he says at the corresponding spoton page 749:

Ein solches hinreichend scharfes Existenztheorem hat inzwischen Eng-ström [1] bewiesen. Auch ergibt sich ein solches, wohl in größtmöglicherAllgemeinheit, aus der kürzlich erschienenen Dissertation von Grunwald[1]; siehe Grunwald [2].

24The footnote continues to announce that this existence theorem will be another one in a series of formerexistence theorems proved by Hasse – same remark as we had already seen above in the paper [AH32].


Such a suffiently strong existence theorem has been proved recently byEngström [1]. Alternatively, it is possible to deduce such a theorem,probably in its greatest possible generality, from the thesis of Grunwald[1] which has recently appeared; see Grunwald [2].

Checking the bibliography of [Has33a] we find under “Engstrom [1]” the entry:“Publication in an American journal in preparation.” However we were not able tofind, either in an American journal or elsewhere, any publication of H. T. Engstromwhere this or a similar theorem is proved.

Howard T. Engström was a young American postdoc from Yale who had stayedin Göttingen for the academic year 1931. Through Emmy Noether he got in contactwith Hasse. He had helped translating Hasse’sAmerican paper [Has32c] into English.Emmy Noether wrote about him in a letter of June 2, 1931:

Engström war mit Ihrem Englisch, bis auf die Umstellungen, sehr zufrie-den; hoffentlich werden Sie es auch mit seinem Existenztheorem seinkönnen! Er ist überhaupt sehr begeistert von allem, was er in Deutschlandgelernt hat. Ich schicke Engströms Manuskript an Deuring, der schonlange ungeduldig darauf ist …

Engström was quite satisfied with your English, apart from the rear-rangements; hopefully you will be satisfied with his existence theoremstoo! He is really very enthusiastic about everything which he had learnedin Germany. I am sending Engström’s manuscript to Deuring who for along time is waiting impatiently for it …

It appears that Hasse had proposed to Engström to write up the proof of the ExistenceTheorem according to his (Hasse’s) outline, and that Deuring was to check Engstrom’smanuscript.

But Engström did not complete his manuscript before he returned to Yale. Wehave found a letter from Engström to Hasse, dated February 27, 1932 from Yale,where he apologizes that he has not finished the manuscript on existence theorems asyet. He concludes:

Your outline indicates to me that you have expended considerable thoughton the matter, and that it would really require not much effort on yourpart to write it up for publication. If this is the case please don’t hesitateto do so ….

We get the impression that Deuring had found a flaw in Engström’s manuscriptand that Hasse had given Engström some hints how to overcome the difficulty. Butat Yale Engström was absorbed by different duties and, hence, returned the subject toHasse.

It remains to check Grunwald, the second reference which was mentioned byHasse in [Has33a].


1.5.3 The Grunwald–Wang story

Grunwald had been a Ph.D. student with Hasse at the University of Halle, and hehad followed Hasse to Marburg in 1930. The reference “Grunwald [1]” in Hasse’spaper refers to Grunwald’s thesis [Gru32] which appeared 1932 in the MathematischeAnnalen. The subject of the thesis belongs to the fundamentals of algebraic numbertheory; from today’s viewpoint it can be viewed as a first attempt to understand therole of Hecke’s Größencharaktere in class field theory. Grunwald’s thesis does notcontain the Existence Theorem, but Hasse discovered that Grunwald’s methods couldbe used to obtain a proof of the theorem. From the correspondence between Grunwaldand Hasse (which is preserved) we can infer that Hasse had proposed to Grunwald toextract from his thesis a proof of the Existence Theorem and publish it in a separatepaper.

And Grunwald did so. The reference “Grunwald [2]” in [Has33a] refers to Grun-wald’s second paper, at that time still “forthcoming”, which appeared 1933 in Crelle’sJournal [Gru33]. There Grunwald proved a general existence theorem which becameknown as “Grunwald’s theorem”. This theorem is much stronger than Hasse’s Exis-tence Theorem:

Grunwald’s theorem. Let K be an algebraic number field and S afinite set of primes of K. For each p 2 S let there be given a cyclicfield extension LpjKp. Moreover, let n 2 N be a common multiple ofthe degrees ŒLp W Kp�. Then there exists a cyclic field extension LjKof degree n such that for each p 2 S its completion coincides with thegiven fields Lp.

Whereas Hasse needed only the fact that the local degrees ŒLp W Kp� should bemultiples of the given numbersmp (for p 2 S ), Grunwald’s theorem claims that eventhe local fields Lp themselves can be prescribed as cyclic extensions of degreemp ofKp (for the finitely many primes p 2 S ). This was a beautiful and strong theorem,and clearly it settled the question.25

The proof of Grunwald’s theorem used class field theory and was considered tobe quite difficult. In 1942 a simplified proof was given by Whaples [Wha42]; it alsoused class field theory but no analytic number theory which had still been necessaryat the time of Grunwald.

In the year 1948 Artin, who was at Princeton University at that time, conducted aseminar on class field theory. One of the seminar talks was devoted to Whaples’ newproof of Grunwald’s theorem. Here is what happened in the seminar, told by one ofthe participants, John Tate26:

25Wilhelm Grunwald (1909–1989) did not continue to work in Mathematics but decided to become a sciencelibrarian. He finally advanced to the position of director of the renowned Göttingen University Library but healways preserved his love for Mathematics, in particular Number Theory. He kept contact with Hasse throughouthis life.

26in a personal letter to the author.


I had just switched from physics to math, and tried to follow it [theseminar] as best I could. Wang also attended that seminar. In the springof 1948, Bill Mills, one of the students Artin had brought with him fromIndiana, talked on “Grunwald’s Theorem” in the seminar. Some dayslater I was with Artin in his office when Wang appeared. He said hehad a counterexample to a lemma which had been used in the proof.An hour or two later, he produced a counterexample to the Theoremitself … Of course he [Artin] was astonished, as were all of us students,that a famous theorem with two published proofs, one of which we hadall heard in the seminar without our noticing anything, could be wrong.But it was a good lesson!

The error was not contained in Grunwald’s paper [Gru33] itself but in Grunwald’sthesis [Gru32] from which the author cited a lemma. That lemma referred to a primenumber p but the author did not see that the prime p D 2 needed special care whencompared with the odd primes p > 2.

The fact that there was an error in Grunwald’s (as well as in Whaples’) theoremcaused a great stir among the people concerned. Would this mean that the MainTheorem of Brauer–Hasse–Noether was wrong too?

Fortunately, the situation was not that serious. In “most” cases Grunwald’s theo-rem holds, and exceptions can only occur ifn is divisible by8. Also, Hasse’s ExistenceTheorem is much weaker than Grunwald’s and it turned out that this weaker theoremholds in any case, including the cases where the full Grunwald theorem collapses.This was established by Wang in his Ph. D. thesis [Wan50], where the whole situationwas investigated and precise conditions were given for the validity of Grunwald’s the-orem. Since then the corrected theorem is called the “Grunwald–Wang Theorem”.27

And the Main Theorem of Brauer–Hasse–Noether was saved.Independently of Wang and immediately after Wang’s counter example became

known, Hasse also published a paper in which he carefully analyzed the exceptionsin Grunwald’s theorem [Has50a]. In particular he too established the validity of hisweaker Existence Theorem which he had used in the Main Theorem.28

In the Artin–Tate Lecture Notes on class field theory of 1951 there is a wholechapter devoted to the Grunwald–Wang theorem [AT68].29

27Shianghao Wang (1915–1993) received his Ph.D. at Princeton University in 1949 and afterwards returned toChina. He published two more papers connected with the Grunwald–Wang theorem but later turned to ComputerScience, in particular control theory. He was professor and chairman of the Math. Dept. at Jiling Universiysince 1952; vice president 1980/81. He became a member of the Academia Sinica. It is said that “Wang was aversatile person. He was good at chess, bridge, novels, Chinese opera.” – I am indebted to Professors Eng TjioeTan and Ming-chang Kang for information about the vita of Shianghao Wang.

28Much later, in the year 2008, Patrick Morton discovered an error in Hasse’s paper which he corrected in[Mor11].

29The statement of theorem 6. chapter 10 in [AT68] contains an essential misprint. This has been remarked byGeyer and Jensen in [GJ07] who also showed that this had already led to erroneous statements in the literature.


1.5.4 Remarks

Let us add some more remarks to this story. Even before the error in Grunwald’stheorem was found, Hasse seemed not to be satisfied with its proof. In his opinionthe proof as given by Grunwald, which used a lot of class field theory, was not adequateas a basis for such a fundamental result like the Main Theorem. Therefore he thoughtof ways to avoid the application of Grunwald’s theorem in the context of the MainTheorem, if possible.

The weak Existence Theorem. The Existence Theorem can be weakened by re-moving the requirement for LjK to have a fixed degree n. In this weak form oneis looking for a cyclic extension LjK whose degree is not specified, with the onlycondition that its local degrees ŒLp W Kp� should be divisible by the given numbersmp (for p 2 S ). Already in 1932 Hasse had given a relatively elementary proof ofthis weak form [Has33a]. This does not yield the full Main Theorem but only itsweak form that every central simple algebra A is similar to a cyclic algebra (whicha priori does not necessarily imply A itself to be cyclic). Quite often this weak formof the Main Theorem turns out to be sufficient in the applications, and so Hasse’sproof in [Has33a] provides a simplified access to those applications, without usingthe complicated class field proof via the Grunwald–Wang theorem.

Moreover, in order to satisfy this weak form of Hasse’s Existence Theorem, itturns out that the required cyclic extension LjK can be constructed as a cyclotomicextension ofK, i.e., a subfield ofK.

p1/ for suitable ` (which may even be chosen as

a prime number).30 This fact became important when Hasse gave a proof of Artin’sReciprocity Law within the framework of the theory of algebras, as he did in [Has33a].See Section 1.6.2.

Group representations. In the joint Brauer–Hasse–Noether paper there is a sectiontitled “Applications” and it is attributed to Hasse. One of those applications concernsrepresentations of finite groups:

Every absolutely irreducible matrix representation of a finite group Gcan be realized (up to equivalence) in the field of nh roots of unity wheren is the order of G and h is sufficiently large.

It is understood that representations are to be meant over a field of characteristic 0. 31

Two representations over the same field are equivalent if they can be transformed intoeach other by a non-singular matrix; in “modern” terms: if they determine isomorphicG-modules.

30The proof needed a lemma on prime numbers satisfying certain congruence relations, of similar kind as Artinhad to use for his general reciprocity law. Simplified proofs of this lemma were later given by Hasse himself, byChevalley, Iyanaga and finally in greater generality by van der Waerden [vdW34].

31At that time the theory of “modular” representations, i.e., over fields of characteristic p > 0, had not yetbeen developed. It had been systematically started by Richard Brauer in the late 1930s.


This theorem had been conjectured by I. Schur [Sch06] but with h D 1. In viewof this Hasse did not say that his theorem is a proof of Schur’s conjecture; instead hesays that this theorem constitutes a “support” of Schur’s conjecture.

In order to prove the theorem, let K D Q. np1/. Hasse considered the group

algebra KŒG� D ˚ Pi Ai , decomposed into its simple components.32 The center

of Ai is K. The assertion of the above theorem is now transformed to say that

K.nhp1/ is a splitting field of each Ai , provided h is sufficiently large. Applying the

Global Splitting Criterion this means that for each prime p of K the p-local degree

of K. nhp1/ over K is a multiple of the local index mi;p. Only the prime divisors p

of n are relevant and for those, mi;p is seen to divide the group order n. Thus one

has to prove the lemma that n divides the local degree K. nhp1/ for each pjn and h

sufficiently large. This then is easily checked by the known decomposition behaviorof primes in cyclotomic fields.33

When sending his draft of their joint manuscript to Brauer on November 11, 1931,Hasse wrote:

Ich könnte mir denken, daß Sie von sich aus zu dem Satz noch etwashinzuzufügen oder eine Verschärfung anzubringen haben. Ich habe dasnur sehr roh angepackt … Habe ich I. Schur richtig und genügend zitiert ?

I could imagine that you would perhaps have a comment to this theoremor maybe a sharper result. I have considered the question only veryroughly … Have I cited I. Schur correctly?

Brauer replied on November 13:

Dieser Satz hat mir besonderen Eindruck gemacht; ich hätte nicht ge-glaubt, daß man die Methoden auf dies Problem so unmittelbar würdeanwenden können. Sehr interessant wäre es ja, wenn man die Zuläs-sigkeit von h D 1 zeigen könnte. Ich bin zur Zeit außerstande, möchtemich aber damit noch weiter beschäftigen, sobald mir der Semester-Anfangstrubel wieder Ruhe dazu läßt – natürlich nur in dem Fall, daß Siees nicht inzwischen selbst erledigt haben, was ich für recht wahrschein-lich halte.

I was particularly impressed by this theorem; I did not believe that onecould apply the methods that directly to this problem. It would be veryinteresting if one could prove h D 1. At present I cannot do this but Iwill think about it as soon as the commotion of the semester beginningis over – of course, only in case that you have not solved the question inthe meantime which I believe is quite probable.

32˚ Pis our notation for direct sum.

33 Actually, the proof in [BHN32] of the last mentioned lemma contains an error. Hasse corrects this error in[Has50b].


Well, Hasse did not solve the question but it was Brauer who many years later verifiedthe Schur conjecture [Bra45]. Two more years later he proved even the stronger resultthat the field of d -th roots of unity suffices where d is the exponent of the group G,[Bra47]. See also [Roq52] and [BT55].

This may be an appropriate occasion to cite a letter which Carl Ludwig Siegelwrote to Hasse on December 7, 1931 when he had been informed about the Brauer–Hasse–Noether paper. It seems that in the past Siegel too had tried to prove Schur’sconjecture, but without success.

Lieber Herr Hasse! … Das ist in der Tat das schönste Geburtstagsge-schenk für Hensel, dass seiner p-adischen Methode ein solcher Tri-umph beschieden wurde. Ich konnte noch nicht einmal das SchurscheProblem richtig anpacken … Der Pessimismus, den ich den Aussichtender Mathematik gegenüber im Allgemeinen empfinde, ist wieder einmalwankend geworden.

Dear Mr. Hasse! … This is indeed the nicest birthday gift for Hensel thathis p-adic methods have been developed to such triumph. I had not evenbeen able to approach the Schur problem properly … The pessimismwhich I harbor generally towards the prospects of mathematics has againbeen shaken …

There had been an exchange of letters between Hasse and Siegel before this. When inJune 1931 Siegel visited Marburg, Hasse told him about his attempts, unsuccessful atthat time, to prove the Local-Global Principle for algebras. On his return to Frankfurt,Siegel wrote a postcard to Hasse with a proof that the discriminant of any divisionalgebra D over a number field is of absolute value > 1; this would have settled theproblem at least if D is central over Q. But after examining Siegel’s proof Hassepointed out to him that this proof does not work, which Siegel conceded (“Manythanks for your exposition of my unsuccessful proof!”).

Algebras with pure maximal subfields. In 1934 there appeared a paper by Albertwith the title “Kummer fields” [Alb34]. There, Albert proved the following theorem:

A central division algebra D of prime degree p over a field K of char-acteristic 0 is cyclic if and only if D contains an element x … K suchthat xp 2 K.34

Of course, this is trivial ifK contains thep-th roots of unity because thenK.x/wouldbe a cyclic subfield ofD of degreep. IfK does not contain thep-th roots of unity thenAlbert constructs a cyclic field LjK contained in D such that L. p

p1/ D K.x;

pp1/;

this can be done by the classical methods of Kummer. Albert formulated this theorem

34Albert calls K.x/ a “pure” extension of K since it is generated by radicals.


over fields of characteristic 0 only, but from its proof it was immediately clear that itremains true over all fields of characteristic ¤ p. 35

Concerning this theorem, Hasse wrote to Albert in February 1935:

Your result seems to me of particular interest. It allows one to eliminateGrunwald’s complicated existence theorem in the proof that every centraldivision algebraD of prime degree p over an algebraic number fieldKis cyclic.

And Hasse proceeds to explain how to derive the Main Theorem for division algebrasof degree p from Albert’s result. This is easy enough. For, let S denote the set ofthose primes p ofK for which the local indexmp ofD is ¤ 1. Choose � 2 K whichis a prime element for every finite p 2 S , and � < 0 for every infinite real p 2 S .Then K. p

p�/ splits D by the Splitting Theorem, hence K. p

p�/ is isomorphic to a

subfield ofD. ApplyingAlbert’s theorem it follows thatD is cyclic. Hasse continues:

We are trying to generalize your theorem to prime power degree. Thiswould eliminate Grunwald’s theorem altogether for the proof of the MainTheorem.

We do not know whether Albert replied to this letter of Hasse. But three years later in[Alb38a] he showed that Hasse’s idea could not be realized. He presented an exampleof a non-cyclic division algebra of index 4 containing a pure subfield of degree 4.The base field K is the rational function field in three variables over a formally realfield.

Exponent = Index. One of the important consequences of the Main Theorem isthe fact that over number fields, the exponent of a central simple algebra equals itsindex. This is a very remarkable theorem. It has interesting consequences in therepresentation theory of finite groups, and this was the reason why Richard Brauerwas particularly interested in it. The theorem does not hold over arbitrary fields sinceBrauer [Bra33] has shown that over function fields of sufficiently many variables,there are division algebras whose exponent e and index m are arbitrarily prescribed,subject only to the conditions which are given in Brauer’s theorems which we havecited in Section 1.4.2 (p. 13). See also Albert [Alb32b] where a similar question isstudied.36

In the Brauer–Hasse–Noether paper the exponent-index theorem is obtained byusing the Existence Theorem (see Section 1.5.1, p. 22). To this end the number n in

35In Hasse’s mathematical diary, dated February 1935 we find the following entry: “Proof of a theorem ofAlbert, following E. Witt.” Witt’s proof of Albert’s theorem is particularly simple, following the style of EmmyNoether, but the essential ingredients are the same as in Albert’s proof. It seems that Witt had presented thisproof in the Hasse seminar, and that Hasse had noted it in his diary for future reference.

36From the references in those papers it appears that Brauer and Albert did not know the results of each otherconcerning this question.


the Existence Theorem is chosen as the least common multiple of the local indicesmp of the p-components Ap. Then L splits A by the Splitting Theorem and hence nis a multiple of the index m of A, hence also of its exponent e. On the other hand,from Hasse’s local theory [Has31d] it follows that the local indexmp equals the localexponent ep of Ap. From Ae � 1 it follows Ae

p � 1 for each p; therefore e is amultiple of ep (for all p) and therefore also of n. It follows e D n D m.

From the above sequence of arguments it is immediate that in fact it is not necessaryto know that LjK is cyclic. But if cyclicity is not required then it is easy, by meansof the Chinese remainder theorem37, to construct a field extension LjK of degreen (as above) with the given local degrees mp for finitely many primes p. This waspointed out by Hasse in his letter to Albert of February 1935 which we had alreadycited above. Hasse wrote:

In my Annalen paper [Has33a] I derived theorem (6.43) (exponent =index) from Grunwald’s existence theorem. In point of fact this deepexistence theorem is not necessary for proving index = exponent. Forone can carry through the proof with any sort of splitting fieldL insteadof a cyclic L. See my first existence theorem [Has26c].38

Grunwald–Wang in the setting of valuation theory. Both proofs of the Grunwald–Wang Theorem, the proof [Wan50] by Wang himself and Hasse’s proof [Has50a], useheavy machinery of class field theory. The same is true withArtin’s proof in theArtin-Tate lecture notes [AT68] where there is a whole chapter devoted to the Grunwald–Wang theorem. But the question arises whether the Grunwald–Wang theorem doesreally belong to class field theory, or perhaps it is valid in a more general setting, forarbitrary fields with valuations. If so then it is to be expected that the proof wouldbecome simpler and more adequate. Therefore Hasse [Has50a] wondered whether itwould be possible to give an algebraic proof using Kummer theory instead of classfield theory. This is indeed possible and has been shown by the author in collaborationwith Falko Lorenz in [LR03]. See also the literature cited there, in particular the paper[Sal82] by Saltmanwho works with generic polynomials.

1.6 The Brauer group and class field theory

The Main Theorem allows us to determine completely the structure of the Brauergroup Br.K/ of a number field K. As we have reported in Section 1.2 already,

37Since there are also infinite primes p involved, the “Chinese remainder theorem” has to be interpreted suchas to include infinite primes too. In other words: this is the theorem of independence of finitely many valuations.

38Hasse’s first existence theorem is stated and proved for finite primes only, i.e., prime ideals in the base field.It seems that Hasse himself, when he cites his paper in the letter to Albert, regarded the inclusion of infiniteprimes in his first existence theorem as trivial (which it is).


the authors of the Brauer–Hasse–Noether paper regard this as one of the importantapplications of the Main Theorem.

Let K be a number field and p a prime of K. If we associate to every centralsimple algebra A over K its completion Ap then we obtain the p-adic localizationmap of Brauer groups Br.K/ ! Br.Kp/. Combining these maps for all primes p ofK we obtain the universal localization map

Br.K/ �! ˚X

p

Br.Kp/

where the sum on the right hand side is understood to be the direct sum. (In thiscontext the Brauer group is written additively.) The Local-Global Principle canbe interpreted to say that this localization map is injective. Thus Br.K/ can beviewed as a subgroup of the direct sum of the local groups Br.Kp/. Accordingly, thedetermination of the structure of Br.K/ starts with the determination of the structureof the local components Br.Kp/.

1.6.1 The local Hasse invariant

First we consider the case when p is a finite prime of K.The description of Br.Kp/ had essentially been done in a former paper by Hasse

[Has31d] in the Mathematische Annalen. We have already had occasion to mentionthis paper in Section 1.5.1 when we discussed the Local Splitting Criterion. Infact, that criterion is a consequence of the following Local Structure Theorem from[Has31d].

We denote by K.n/p the unramified extension of Kp of degree n. It is cyclic, and

the Galois group is generated by the Frobenius automorphism; let us call it '. 39 In[Has31d] we find the following

Local Structure Theorem (level n). LetAp be any central simple alge-bra overKp, of dimensionn2. ThenAp contains amaximal commutative

subfield isomorphic to K.n/p . Consequently Ap is cyclic and admits a

representation of the form Ap D .K.n/p ; '; a/ with a 2 K�

p .

The remarkable fact is not only that all of those algebras Ap are cyclic, but that

each of them contains the same canonical field extension K.n/p as a maximal cyclic

subfield. Even more remarkable is how Hasse had derived this. Namely, he appliedthe classical p-adic methods of Hensel to the non-commutative case. Let us explainthis:

39More precisely, we should perhaps write '.n/ since it is an automorphism of K.n/p depending on n. But let

us interpret the symbol ' as the Frobenius automorphism of the maximal unramified extension of Kp; if applied

to the elements of K.n/p this gives the Frobenius automorphism of K

.n/p . This simplifies the notation somewhat.


The general case is readily reduced to the case whenAp D Dp is a division algebraof dimension n2 over Kp. Now, Kp being a complete field, it carries canonically avaluation which we denote by v. Writing this valuation additively, the axioms for thevaluation are

v.ab/ D v.a/C v.b/;

v.aC b/ � min.v.a/; v.b//:

Now, Hasse’s method consisted of extending this valuation to the given divisionalgebra Dp. It turns out that such an extension is uniquely possible; the formula forthe extended valuation is

v.x/ D 1

nv.Nx/ .x 2 Dp/;

where N denotes the reduced norm from Dp to Kp. This formula and the proof areprecisely the same as developed by Hensel for extending valuations to commutativeextensions, in particular it uses Hensel’s Lemma. Now Dp appears as a valued skewfield with center Kp. As such it has a ramification degree e and a residue degree f .But unlike the commutative case it turns out that here, Kp being the center of Dp,we have always e D f D n. Since f D n it follows readily from Hensel’s Lemmathat Dp contains the unramified field extension K.n/

p of degree n, as announced inthe theorem.

We have said above that this proof is remarkable. This does not mean that theproof is difficult; in fact, it is straightforward for anyone who is acquainted withHensel’s method of handling valuations. The remarkable thing is that Hasse usedvaluations to investigate non-commutative division algebras over local fields.40 Thevaluation ring of Dp consists of all x 2 Dp with v.x/ � 0. It contains a uniquemaximal ideal, which is a 2-sided ideal, consisting of all x with v.x/ > 0.

Remark. Hasse denotes this valuation prime ideal by the letter }, and this shows upin the title of his paper [Has31d]. This somewhat strange notation is explained bythe fact that, in Hasse’s paper, the symbol p is used for the canonical prime ideal inthe complete field Kp, and the corresponding capital letter P was used to denote thevaluation ideal in commutative field extensions. Thus, in order to indicate that in thenon-commutative case the situation is somewhat different, Hasse proposed to use adifferent symbol, and he chose } for this purpose. Formerly this symbol, known asthe Weierstrass-p, was used to denote the elliptic function }.z/ in the Weierstrassnormalization. Hasse’s notation for the prime ideal of a valued division algebra didnot survive, but the Weierstrass notation }.z/ is still in use today in the theory ofelliptic functions.

40After Hasse, the valuation theory of non-commutative structures developed rapidly, not only over numberfields but over arbitrary fields. We refer to the impressive report of Wadsworth [Wad02] about this development.All this started with Hasse’s paper [Has31d] which is under discussion here.


Emmy Noether, after having read Hasse’s manuscript for [Has31d], recognizedimmediately its strong potential. Hasse had sent this manuscript to her for publicationin the Mathematische Annalen for which Noether acted as unofficial editor.41 On apostcard dated June 25, 1930 she wrote to Hasse:

Lieber Herr Hasse! Ihre hyperkomplexe p-adik hat mir sehr viel Freudegemacht.

Dear Mr. Hasse! I have found your paper on hypercomplex p-adics veryenjoyable.42

And, as it was her custom, she immediately jotted down her comments and proposalsfor further studies. About the local theory, which concerns us here, she wrote:

… Aus der Klassenkörpertheorie im Kleinen folgt: ist Lp zyklischn-ten Grades über einem p-adischen Grundkörper Kp, so gibt es inKp wenigstens ein Element a ¤ 0, derart, dass erst an Norm einesLp-Elements wird. Können Sie das direkt beweisen? Dann könnte manaus Ihren Schiefkörperergebnissen umgekehrt die Klassenkörpertheorieim Kleinen begründen …

… From local class field theory it follows: If Lp is cyclic of degree nover a p-adic base field Kp then there exists at least one element a ¤ 0

in Kp such that only an becomes a norm of an Lp-element. Are youable to prove this directly? Then one could derive local class field theoryfrom your skew field results …

In other words: Noether asks whether the Brauer group Br.LpjKp/ contains anelement of exponent n.

We do not know precisely what Hasse replied to her. But from later correspon-dence with Noether we can implicitly conclude that he replied something like “I donot know”. It took him some time to follow her hint and to realize that he could havesaid “yes” in view of the Local Structure Theorem above.

Let us briefly indicate the arguments which Hasse could have used. These can befound in Hasse’s later papers, the Brauer–Hasse–Noether paper [BHN32], the Amer-ican paper [Has32c] and the paper [Has33a] dedicated to Noether’s 50th birthday.But in fact the arguments are essentially based on the Local Structure Theorem.

If we associate to each a 2 K�p the cyclic algebra .K.n/

p ; '; a/ then we obtain a

homomorphism from K�p to the Brauer group Br.K.n/

p jKp/. By the Local Structure

41“Unofficial” means that her name was not mentioned officially on the title page. But people who knew senttheir paper to her if it belonged to Noether’s field of interest. As a rule, Noether read the paper and, if she foundit suitable, sent it to Blumenthal who, as the managing editor, accepted it. The date of “received by the editors”was set as the date when the paper was received by Emmy Noether. Thus Hasse’s paper [Has31d] carries thedate of June 16, 1930.

42Emmy Noether used the symbol p (in German handwriting) since she did not know how to write } as sheadmitted in a later letter. Can we conclude from this that she never had worked with elliptic functions?


Theorem this homomorphism is surjective. Its kernel is the group of norms fromK

.n/p . But K.n/

p is unramified over Kp and therefore an element a 2 K�p is a norm

if and only if its value v.a/ 0 mod n. 43 Consequently, for any a 2 K�p its value

v.a/ modulo n represents its class in the norm class group, and hence represents thealgebra .K.n/

p ; '; a/. In particular we see that Br.K.n/p jKp/ is isomorphic to Z=n.

By the Local Structure Theorem, Br.K.n/p jKp/ contains all central simple algebras

Ap of index dividing n. Consequently, ifLp is an arbitrary extension ofKp of degreeŒLp W Kp� D n then the Brauer group

Br.LpjKp/ � Br.K.n/p jKp/ � Z=n:

Hence, Noether’s question is answered affirmatively if we have equality here, whichis to say that every algebra Ap in Br.K.n/

p jKp/ is split by Lp. Since the index of Ap

divides n D ŒLp W Kp� this follows from the Local Splitting Criterion.Thus indeed, Hasse could have answered Noether’s question with “yes”, already in

1930 ; in fact he did so later. But Noether’s conclusion that one could derive local classfield theory from this, was too optimistic. Noether’s question was concerned withcyclic extensions only, but class field theory deals with arbitrary abelian extensions.It was only later that Chevalley [Che33] showed how to perform the transition fromcyclic to arbitrary abelian extensions.

Let us return to the Local Structure Theorem for level n. We have seen abovethat this implies an isomorphism Br.K.n/

p jKp/ � Z=n, via the map .K.n/p ; '; a/ 7!

v.a/ mod n. In other words: the residue class v.a/ mod n is an invariant of thealgebra Ap D .K

.n/p ; '; a/. To obtain an invariant which is independent of n Hasse

divided v.a/ by n and thus defined what today is called the Hasse-invariant:

�Ap

p

�W v.a/

nmod Z I

this is a certain rational number which is determined modulo integers only. If m isa multiple of n then every Ap 2 Br.K.n/

p jKp/ can also be viewed to be contained in

Br.K.m/p jKp/ and it turns out that the Hasse invariant as defined above is the same

for m as that for n. If n ! 1 the final version of the Local Structure Theorem forthe Brauer group emerges:

Local Structure Theorem. Let p be a finite prime ofK. If we associateto every central simple algebra over Kp its Hasse invariant then weobtain the canonical group isomorphism

invp W Br.Kp/ ��!� Q=Z:

43We assume here that the valuation of Kp is normalized with value group v.K�p / D Z.


If p is an infinite real prime thenKp D R and there is only one non-trivial centraldivision algebra over R, namely the quaternions of Hamilton. If we associate to thisthe Hasse-invariant 1

2then we obtain

invp W Br.Kp/ ��!�´

12Z=Z if p is real,

0 if p is complex.

1.6.2 Structure of the global Brauer group

Having settled the local structure theorems, Hasse turns now to the global structure,i.e., the structure of Br.K/ for a number fieldK. IfA is a central simple algebra over

K and Ap its p-adic completion then Hasse writes briefly�

Ap

�instead of

�Ap

p

�. If we

associate to each A its local Hasse invariants�

Ap

�then we obtain the global invariant

map

inv W Br.K/ �! ˚X 0

p

Q=Z

where the prime on the † sign of the direct sum should remind the reader that if p isinfinite then Q=Z has to be replaced by 1

2Z=Z or 0 according to whether p is real

or complex. By the Local-Global Principle this global invariant map is injective. Inother words: every algebraA in Br.K/ is uniquely determined by its Hasse invariants(up to similarity). In order to describe the structure of Br.K/ completely one has todescribe the image of the invariant map. In other words: What are the conditions thata given system of rational numbers, rp for each prime p ofK, is the system of Hasseinvariants of some algebra A 2 Br.K/ ? The following conditions are evident:

1. There are only finitely many p with rp 6 0 mod Z.

2. For an infinite p, we have rp ´0 or 1

2mod Z if p is real,

0 mod Z if p is complex.

Apart from this there is only one further condition, expressed in the following structuretheorem for the global Brauer group.

Global Structure Theorem. (i) For any central simple algebra A overthe number field K, the sum formula for its Hasse invariants holds:

Xp

�A

p

� 0 mod Z:

(ii) If rp is an arbitrary system of rational numbers, subject to the condi-tions 1. and 2. above, and if

Pp rp 0 mod Z, then there is a unique

A 2 Br.K/ such that�

Ap

� rp mod Z for each p.


Today we would express this theorem by saying that the sequence of canonical maps

0 ! Br.K/inv��! ˚

X 0

p

Q=Zadd��! Q=Z ! 0 (1.1)

is exact. This describes the Brauer group Br.K/, if considered via the map “inv” asa subgroup of the direct sum, as being the kernel of the map “add” which adds thecomponents of the direct sum.

Although the Global Structure Theorem uses the Local-Global Principle and isbuilt on it, it is by no means an easy consequence of it. Perhaps this is the reason whythe theorem is not treated in the Brauer–Hasse–Noether paper [BHN32]; recall thatthis paper had to be written in haste. Nevertheless, as said above already, the authorsof [BHN32] stressed the point that their Local-Global Principle is of fundamentalimportance for the structure of the Brauer group.

The Global Structure Theorem, at least its first part (i), is in fact equivalent toArtin’s General Reciprocity Law of class field theory. Hasse’s proof can be foundin [Has33a], and it is the end point of a historic string of events stretching overseveral years since 1927. Let us briefly sketch chronologically the highlights in thisdevelopment.

1927. Artin succeeded in [Art27] to prove his General Reciprocity Law which he hadconjectured since 1923. Given an abelian extension LjK of number fields, Artin’stheorem established an isomorphism between the group of divisor classes attachedto LjK in the sense of class field theory, and the Galois group G of LjK. Thisisomorphism is obtained by associating to every prime p of K which is unramifiedin L, its Frobenius automorphism 'p 2 G. This theorem has been said to be “thecoronation of Takagi’s class field theory”.

Even before the appearance of his paper [Art27], Artin informed Hasse about hisresult and its proof. There followed an intense exchange of letters between Hasse andArtin discussing the consequences of Artin’s Reciprocity Law. Already in his firstsuch letter dated July 17, 1927, Artin mentioned that probably Hilbert’s version of thereciprocity law may now be proved in full generality. Later he asked Hasse whetherhe could do it and Hasse agreed. Accordingly, Hasse published in the same year 1927a supplement to Artin’s Reciprocity Law [Has27c] where (among other things) theproduct formula for the general m-th Hilbert symbol

�a; bp

�m

was established.Here we do not intend to describe the definition of the Hilbert symbol; let it be

sufficient to say that it serves to decide whether a given number a 2 K is an m-thpower modulo a prime p, and that the product formula includes as a special case theKummer m-th power reciprocity law together with its various supplementary laws.

The definition and management of the Hilbert symbol requires that them-th rootsof unity are contained in the base fieldK. Now Artin, in another letter to Hasse datedJuly 21, 1927, asked whether it would be possible to define some kind of Hilbert


symbol without assuming that the proper roots of unity are contained in K. Hassesucceeded with this in 1929.

1929. Hasse’s paper [Has30d] appeared in 1930 but since it had been received by theeditors on March 7, 1929 already we count it for 1929. In this paper Hasse definedfor an arbitrary abelian extensionLjK and a 2 K�, for each prime p ofK the “normsymbol”

�a; LjK

p

�as an element of the Galois group of LjK. More precisely, it is

an element of the p-adic decomposition group Gp � G. This symbol assumes thevalue 1 if and only if a is a norm in the local extension LpjKp. The norm symbolis in some sense a generalization of the Hilbert symbol. If p is unramified in L anda D � a prime element for p then

��; LjK

p

�equals the Frobenius automorphism 'p

appearing in the Artin map. Hasse’s norm symbol satisfies the product formula:

Yp

�a; LjK

p

�D 1

where the 1 on the right hand side denotes the neutral element of the Galois group.Hasse obtained this product formula from Artin’s Reciprocity Law. Conversely,Artin’s Reciprocity Law may be deduced from the above product formula.

As a side remark we mention that through this definition of the norm symbolHasse discovered local class field theory. See [Has30c].

1931. On May 29, 1931 Hasse submitted his American paper [Has32c] to the Trans-actions of the AMS. In that paper he presented a comprehensive treatment of cyclicalgebras over number fields. This was prior to the discovery of the Main Theorem,so Hasse did not yet know that every central simple algebra over a number field Kis cyclic. For a cyclic algebra A D .LjK;S; a/ he compared the Hasse invariant�

Ap

�with the norm symbol

�a;LjK

p

�. It turned out that the product formula of the

(multiplicative) norm symbol provides the key for the proof of the sum formula forthe (additive) Hasse invariant as stated in part (i) of the Global Structure Theorem.

Actually, the sum formula was not yet explicitly written down in Hasse’s Amer-ican paper; this was done in his next paper [Has33a] only. But all the necessaryingredients and computations can be found in Hasse’s American paper [Has32c] al-ready. Although that paper had not yet appeared when the Brauer–Hasse–Noetherpaper [BHN32] was written, the content of Hasse’s American paper was known toBrauer and Noether too since Hasse had informed them about his results.

1932. In March 1932 Hasse sent his dedication paper [Has33a] to Emmy Noether.This paper does not only contain a new proof arrangement for the Main Theorem, aswe had reported earlier. In addition, Hasse stated and proved explicitly the GlobalStructure Theorem, not only Part (i) (which tacitly was already contained in [Has32c])but also Part (ii). Moreover, the paper went well beyond Hasse’s former papers in asmuch as now he did not use Artin’s Reciprocity Law in proving the Global StructureTheorem.


We note that all three foregoing papers of Hasse, those of 1927, 1929 and 1931,were built on Artin’s Reciprocity Law because the definition of the local norm symbol�

a; LjKp

�depended on Artin’s global law. But now, in the Noether dedication paper,

Hasse was able to use his invariants�

Ap

�for a purely local definition of the norm

symbol�

a; LjKp

�. This had been suggested to him by Emmy Noether who on a postcard

of April 12, 1931 wrote the following. This letter was the reaction of Noether to areport of Hasse about his results in his American paper [Has32c].

Ihre Sätze habe ich mit großer Begeisterung, wie einen spannenden Ro-man gelesen; Sie sind wirklich weit gekommen! Jetzt … wünsche ichmir noch die Umkehrung: direkte hyperkomplexe Begründung der In-varianten … und damit hyperkomplexe Begründung des Reziprozitäts-gesetzes! Aber das hat wohl noch gute Weile! Immerhin haben Sie doch,wenn ich mich recht erinnere, in der Schiefkörper-Arbeit mit Ihren Ex-ponenten ep den ersten Teil schon gemacht?

I have read your theorems with great enthusiasm, like a thrilling novel;you have got really very far! Now … I wish to have also the reverse:direct hypercomplex foundation of the invariants … and thus hypercom-plex foundation of the reciprocity law! But this may take still sometime! Nevertheless you had done, if I remember correctly, the first stepalready in your skew field paper with the exponents ep?

The “skew field paper” which Noether mentions, is Hasse’s [Has31d] which wehad discussed above in Section 1.6.1.44 As we have explained there, the Local Struc-ture Theorem indeed can be used to provide a local definition of the Hasse invariant.Thus Noether had seen clearly the potential of this for her plan to reverse the argu-ment, so that one first proves the sum formula of the Global Structure Theorem, and

then interpret this as the product formula for Hasse’s norm residue symbol�

a;LjKp

�.

The latter is equivalent to Artin’s Reciprocity Law.And Hasse followed Noether’s hint and succeeded to give what Noether called a

“ hypercomplex proof of Artin’s Reciprocity Law”. Thus a close connection betweenthe theory of algebras and class field theory became visible.

WhileArtin’s paper [Art27] with his reciprocity law had been named as the “Coro-nation of class field theory”, similarly Hasse’s paper [Has33a] could now be regardedas the “Coronation of the theory of algebras”.

We can now understand Artin’s exclamation which we have cited in Section 1.2,namely that he regards this as the “greatest advance in Number Theory of the lastyears …” When Artin wrote that letter in November 1931, Hasse’s paper [Has33a]had not yet appeared. But Artin seemed to have clearly seen, as Emmy Noether haddone, the potential of the Local-Global Principle as a foundation of class field theory.In fact, in his letter he continued:

44See also Section 1.6.3 below.


Ich lese jetzt Klassenkörpertheorie und will nächstes Semester anschlie-ßend hyperkomplex werden.

At present I am giving a course on class field theory, and next semesterI will continue by becoming hypercomplex.

Thus Artin intended to discuss the theory of hypercomplex systems, i.e., algebras,with the view of its application to class field theory.45

1.6.3 Remarks

Arithmetic of algebras and Hensel’s methods. Twice in our discussion we hadoccasion to refer to Hasse’s paper [Has31d] on local algebras. The first time thiswas in Section 1.5.1 when we reported that the local splitting theorem was an almostimmediate consequence of the results of that paper. The second time was in Sec-tion 1.6.1 when we discussed the local structure theorems. In both situations we haveseen that Hasse’s paper [Has31d] contained the fundamental ingredients which ledto success.

We note that this paper [Has31d] was received by the editors on June 18, 1930already, long before the Brauer–Hasse–Noether paper was composed, and even beforeHasse had formulated the conjecture of the Main Theorem in a letter to Noether (seeSection 1.7.2, p. 51). In fact, the original motivation for Hasse to write this paper wasnot directly connected with the Main Theorem. From the introduction of [Has31d]we infer that Hasse regarded his paper as a new approach to understand the arithmeticof algebras, based on the ideas of Hensel, in the same manner as he had applied thoseideas to the investigation of the arithmetic of commutative number fields.

We cannot here give a comprehensive account of the development of the arithmeticof algebras during the 1920s; this is an exciting story but would need much more spacethan is available. The following brief comments should help to put Hasse’s paper[Has31d] into the right perspective.

The study of the arithmetic theory of algebras had been started systematically byDickson whose book [Dic23], entitled

Algebras and their Arithmetics,

had received much attention, in particular among German mathematicians. This bookcontained not only a complete treatment of the Wedderburn structure theorems foralgebras, but also a systematic attempt to develop an arithmetic theory of orders ofan algebra.

45The “next semester” was the summer term 1932. In that semesterArtin gave a course with the title “Algebra”,and he presented there the algebraic theory of hypercomplex systems, i.e., algebras. Lecture Notes for this coursehad been taken down by the student Ernst August Eichelbrenner, and a copy is preserved. However from thesenotes it appears that Artin covered the algebraic theory of algebras over an arbitrary field only, but not the specialsituation when the base field is a number field. In particular, the connection to class field theory is not mentioned.But it may well have been that Artin covered those more advanced topics in a special seminar parallel to thiscourse.


Let K be a number field and ZK its ring of integers. Let A be any finite-dimensional algebra over K. In this situation an order R is defined to be a subringof A (containing the unit element) which is a finite ZK-module and generates A as avector space overK. The “arithmetic” of R manifests itself in the structure of idealsof R. The arithmetic of such order becomes particularly lucid if the order is maxi-mal, i.e., not properly contained in a larger order of the algebra. Perhaps it is not anexaggeration to say that the most important feature of Dickson’s book was to give thedefinition of maximal orders of an algebra and to point out that the arithmetic of thosemaximal orders is particularly important – in the same way as in the commutativecase, i.e., algebraic number fieldsK, where the maximal order ZK and its prime idealstructure is the first object to study. Whereas arbitrary orders, i.e., those which arenot integrally closed, carry a more complicated ideal theory.

Now we observe that in Dickson’s book, after maximal orders have been definedand their elementary properties developed, they are in fact not treated systematically.The discussion is largely restricted to very special cases, namely when there exists aeuclidean algorithm. One knows in algebraic number theory that such cases are rare.

The first who set out to remedy this unsatisfying situation was Andreas Speiserin his paper [Spe26]. Also, he arranged for a German translation of Dickson’s book[Dic27] and included his paper as an additional chapter.46

But still, Speiser’s treatment was only the beginning. SoonArtin published a seriesof three seminal papers in which the arithmetic of maximal orders was fully developed[Art28a, Art28c, Art28b]. The second of these papers contained the generalizationof Wedderburn’s structure theorems to what today are called “Artinian rings”, i.e.,rings with minimum condition for ideals.47 This is necessary if one wishes to studythe structure of the residue class rings of a maximal order with respect to arbitrarytwo-sided ideals which are not necessarily prime. The third of Artin’s papers thendeveloped the ideal theory of maximal orders of a simple algebra, in complete analogyto Noether’s theory of Dedekind rings (which had just been published the year before).The non-commutativity implies that the ideals do not necessarily form a group but(with proper definition of multiplication) a so-called “groupoid” in the sense of Brandt[Bra30a].

It was Speiser’s work and, in addition, this series of papers by Artin which hadinspired Hasse to write his paper [Has31d] on local division algebras. In his intro-duction he refers to Speiser and Artin, and says:

Ich baue den ursprünglichen Speiserschen Ansatz in demselben Sinneaus, wie es die Henselsche Arithmetik der algebraischen Zahlkörper mit

46The translation had been done by J. J. Burckhardt who recently had his 100th birthday in good health.See [Fre03]. – Actually, the German edition is not merely a translation of the American book. Dickson hadpresented a completely reworked manuscript for translation. The book had been reviewed by Hasse [Has28] inthe Jahresbericht der DMV.

47At that time it was not yet known that the minimum condition for ideals implies the maximum condition.Hence Artin in his paper required the validity of the minimum as well as the maximum condition.


dem ursprünglichen … Ansatz von Kummer tut. Betrachtet man näm-lich … gleichzeitig die Restsysteme nach jeder noch so hohen Potenzvon p, so kommt das darauf hinaus, daß man den rationalen Koeffizien-tenkörper p-adisch erweitert. An Stelle des Speiserschen Restsystemsmod ps tritt demzufolge … ein hyperkomplexes System in bezug aufeinen Körper, den p-adischen Zahlkörper, im Sinne der Wedderburn-schen Theorie.

Auf dieser einfachen Grundlage gelingt es überraschend einfach … denAufbau der hyperkomplexen Arithmetik zu vollziehen.

I am extending the original idea of Speiser in the same sense as Hensel’sarithmetic of number fields had done with the original … idea of Kum-mer. For, if one considers … simultaneously the residue classes withrespect to arbitrary powers of p then this means to extend the field ofcoefficients p-adically. In this way, Speiser’s residue classes modulops … are replaced by an algebra over a field, the p-adic completion, inthe sense of Wedderburn.

In this way it is possible to build the hypercomplex arithmetic in a sur-prisingly simple way.

And as part of this program, Hasse mentions:

Darüber hinaus gelingt es mir, eine einfache Übersicht über alle über-haupt vorhandenen Schiefkörper über einem p-adischen algebraischenZahlkörper als Zentrum und … deren algebraische und arithmetischeStruktur zu erhalten, in Analogie zu der bekannten Tatsache, daß esüber dem reellen Zahlkörper als Zentrum nur einen Schiefkörper, denQuaternionenkörper, gibt.

Moreover I have succeeded in giving a complete description of all exist-ing division algebras with a p-adic algebraic number field as its center,and of its arithmetic and algebraic structure – in analogy to the well-known fact that with the real number field as center there exists only oneskew field, the ordinary quaternion field.

Thus Hasse’s local structure theorems which we had cited from [Has31a], constitutedonly one aspect of this paper. The other and broader one was to build non-commutativearithmetic in maximal orders of algebras, by using Hensel’s ideas of localization.

In this light we can understand why Hasse in his dedication text of the Brauer–Hasse–Noether paper had mentioned “Hensel’sp-adic methods” as being responsiblefor the success.

Class field theory and cohomology. In the section “Consequences” (“Folgerun-gen”) of the Brauer–Hasse–Noether paper there is a subsection concerning


Verallgemeinerung von Hauptsätzen der Klassenkörpertheorie auf all-gemeine relativ-galoissche Zahlkörper.

Generalization of central theorems of class field theory to the case ofarbitrary Galois extensions of number fields.

The problem is the following: Ordinary class field theory in the sense of Takagi refersto abelian extensions of number fields. The abelian extensionsL of a number fieldKare characterized by certain groups, called “ray class groups” of divisors, which areconstructed within the base fieldK – in such a way that the decomposition type in Lof primes p ofK can be read off from the behavior of the primes in the correspondingray class group. Question: Is a characterization of a similar kind possible for Galoisextensions ofK which are not necessarily abelian? It was known since Hasse’s classfield report [Has26a] that this is not possible by means of ray class groups; this is thecontent of what Hasse [Has33b] called “Abgrenzungssatz” (theorem of delimitation).But there may be other groups or objects which are defined within K and can serveto describe Galois extensions LjK.

Now, in the Brauer–Hasse–Noether paper, Hasse proposes to use Brauer groups.He shows that every Galois extension LjK of number fields is uniquely determinedby its Brauer group Br.LjK/. And the decomposition type in L of a prime p of Kcan be read off from the p-adic behavior of the elements in Br.LjK/. The proof isalmost immediate using the splitting theorems of Section 1.5.1 together with wellknown density theorems of algebraic number theory.

In consequence there arises the problem how to describe the Brauer groupsBr.LjK/withinK without resorting toL. This problem is not treated in the Brauer–Hasse–Noether paper. However it has stimulated several mathematicians, includingArtin and Noether, to look more closely into the Brauer group or, equivalently, intothe cohomology group H 2.LjK/.

In theArtin–Hasse correspondence we find 5 letters between March and May 1932where Artin tries to give congruence criteria for the decomposition type of a primep of K in L by means of factor systems. However his results were disappointing tohim. He wrote to Hasse:

Im nicht abelschen Fall kommt einfach die alte Methode heraus dieKlassenkörpertheorie anzuwenden auf Unterkörper in bezug auf dieder ganze Körper cyklisch ist … Dieses …ist nur eine etwas verschönteZusammenfassung … der gewöhnlichen Klassenkörpertheorie … Ichhabe den Eindruck, dass noch etwas ganz Neues hinzukommen mussum zu Isomorphie und zu Existenzsätzen zu kommen.

In the non-abelian case we just obtain the old method to apply class fieldtheory to subfields over which the whole field is cyclic … This is only asomewhat beautified combination of the ordinary class field theory … Ihave the impression that something completely new has to be added.


Hasse seems to have been more optimistic. In his talk [Has32b] at the InternationalCongress of Mathematicians, Zürich 1932, he said, after having reported about theMain Theorem:

Schließlich erweist sich in Untersuchungen von Artin, E. Noether undmir Satz 3 (und überhaupt diese Methode) als kräftiges Hilfsmittelbei der Behandlung der großen, im Mittelpunkt der modernen Zahlen-theorie stehenden Frage nach dem Zerlegungsgesetz in allg. galoisschenZahlkörpern.

Further work of Artin, E. Noether and myself has shown that Theorem 3(and that method in general) is a powerful tool to deal with the greatquestion in the center of modern number theory, the decomposition lawin general Galois number fields.

Here, “Theorem 3” means the Main Theorem, and “that method” had been explainedin Hasse’s text before, namely:

… Kombination der von Hensel geschaffenen arithm. Methoden, dieich im Anschluss an Speiser in diese Theorie hereingetragen habe, mitgewissen algebr. Methoden, die, auf früheren Untersuchungen von Spei-ser und I. Schur fußend, kürzlich von R. Brauer und E. Noether entwick-elt wurden.

… a combination of the arithmetic methods of Hensel, which I havecarried into this theory following Speiser, with certain algebraic meth-ods which, based on earlier investigations of Speiser and I. Schur, haverecently been developed by R. Brauer and E. Noether.48

Emmy Noether seems to have steered a middle line. On the one hand she wasinformed about Artin’s unsuccessful attempts. In her invited address [Noe32] at theZürich Congress she says the following, after having reported on Hasse’s proof of thereciprocity law by means of algebras, and on some further developments of Chevalley[Che33] about factor systems:

Zugleich muss ich aber doch einschränkend bemerken, dass die Methodeder verschränkten Produkte allein allem Anschein nach nicht die volleTheorie des galoisschen Zahlkörpers ergibt. Das folgt aus neuen nochunpublizierten Arbeiten von Artin, die an den obigen Beweis von Hasseanschliessen …

48It may seem strange that Hasse did not mention Albert who also had an independent share in the proof ofthe Main Theorem (see Section 1.8.) We can only speculate about the reason for this (if there was any particularreason at all). It may have been the fact that, for one thing, Hasse had mentioned Albert earlier in his texttogether with Dickson and Wedderburn, and that on the other hand the methods which he refers to in the presentconnection are concerned with computations on factor systems in the realm of class field theory, which indeedcannot be found in Albert’s papers.


At the same time I have to qualify this by saying that the method ofcrossed products does not seem to cover the full theory of Galois numberfields. This is a consequence of new, still unpublished results of Artinwhich are based on Hasse’s above mentioned proof …

On the other hand, Emmy Noether in her own work pushes the computationwith factor systems further along by what she calls the “Principal Genus Theorem”(“Hauptgeschlechtssatz”) [Noe33a]. Seen from today’s viewpoint, her work is ofcohomological nature and her Principal Genus Theorem is essentially the vanishingof the 1-cohomology of the idele class group, or at least of some finite level of it.

In the course of later developments the idea of approaching class field theory forGalois extensions by means of factor systems has been dropped. The new concept foran edifice of class field theory for Galois extensions, due to Langlands, looks quitedifferent.

But the extensive computations with factor systems have had a significant conse-quence in the long term, namely the rise of algebraic cohomology and its applicationin algebraic number theory. While Hasse had introduced simple algebras into classfield theory in [Has33a], these have survived in modern times as 2-cohomologyclasses only. Accordingly the exact sequence which we have written down in (1.1)immediately after the Global Structure Theorem (p. 37), is now written in the form

0 ! H 2.K/�!H 2.IK/�!Q=Z ! 0;

IK being the idele group (suitably topologized) and H 2.IK/ D ˚ PpH

2.K�p /:

Many years later, Artin and Tate presented in their 1952 Seminar Notes [AT68] anaxiomatic foundation of class field theory. Their axioms were given in the language ofcohomology which by then was already well developed. There are two main axioms.Their Axiom I is essentially the cohomological version of the exactness of the abovesequence at the term H 2.K/. And their Axiom II is essentially equivalent to theexactness at H 2.IK/.

We have mentioned all this in order to point out that it started with the Brauer–Hasse–Noether paper [BHN32].

1.7 The team: Noether, Brauer and Hasse

As seen above, there was a close collaboration between Brauer, Hasse and Noetherwhich finally led to the Main Theorem. We have tried to find out how this cooperationstarted and developed.


1.7.1 Noether’s error

We do not know when Emmy Noether and Richard Brauer had met for the first time.There is a letter from Noether to Brauer dated March 28, 1927 which seems to be areply to a previous letter from Brauer to her.49 The letter is published and discussedin the beautiful book of C. Curtis Pioneers of representation theory [Cur99], p. 226;it starts as follows:

Sehr geehrter Herr Brauer! Es freut mich sehr, daß Sie jetzt auch denZusammenhang der Darstellungstheorie und der Theorie der nichtkom-mutativen Ringe, der „Algebren“, erkannt haben; und den Zusammen-hang zwischen Schurschem Index und Divisionsalgebren.

Dear Mr. Brauer! I am very glad that now you have also recognized theconnection between representation theory and the theory of noncommu-tative rings, the “algebras”, and the connection between the Schur indexand division algebras.

The tone of the letter is somewhat like that from an instructor to a young student50,giving him good marks for success in his studies. But then she becomes serious:

In diesen Grundlagen stimmen unsere Untersuchungen natürlich über-ein; aber dann scheint mir ein Auseinandergehen zu sein.

In regard to these fundamentals our investigations are, of course, inagreement; but then it seems to me there is a divergence.

And she continues to describe this divergence, followed by an essay on how she likesto view the situation, with the unspoken invitation that Brauer too should take thesame viewpoint.

The subject is representation theory, the Schur index and splitting fields. Noetheradvocates that the whole theory be subsumed under the theory of algebras. On thisproject she had been working already for some time. In the winter semester 1924/25she had given a course on this subject.51 In September 1925 she had given a talk atthe annual meeting of the DMV at Danzig with the title “Group characters and idealtheory”.52 In the abstract of this talk [Noe25] she writes:

49In the same year 1927 Brauer did his “Habilitation” at the University of Königsberg. It seems that Brauerhad sent his thesis (Habilitationsschrift) to Noether asking for her comments, and that the above mentioned letteris Noether’s reply. Brauer submitted his paper [Bra28] to the Mathematische Zeitschrift on July 22, 1927, fourmonths after Noether’s letter.

50Noether was 19 years older than Brauer, who was 26.51I am indebted to Mrs. Mechthild Koreuber for showing me her list of the Noether lectures in Göttingen 1916–

1933, copied from the “Vorlesungsverzeichnis”. For the winter semester 1924/25 we find the announcement ofa 4-hour lecture on group theory. This seems to be the lecture where Noether first expounded her ideas of doingrepresentation theory within the framework of algebras.

52This was the same meeting where Hasse gave his famous report on class field theory. Noether’s and Hasse’stalks were scheduled at the same session. See [Roq01].


Die Frobeniussche Theorie der Gruppencharaktere – also der Darstel-lung endlicher Gruppen – wird aufgefaßt als Idealtheorie eines voll-ständig reduziblen Ringes, des Gruppenringes.

Frobenius’ theory of group characters – i.e., representation of finitegroups – is seen as the ideal theory of a completely reducible ring, thegroup ring.

She continues with the Wedderburn structure theorems for algebras and how theseare to be interpreted within representation theory. She ends up with the sentences:

Damit ist aber die Frobeniussche Theorie eingeordnet. Eine ausführlicheDarstellung soll in den Math. Ann. erscheinen.

Thus the theory of Frobenius is subsumed. A detailed account is to bepublished in the Math. Ann.

We see that already in 1925 Noether had a clear view of what was necessary todevelop representation theory within the framework of algebras. But the promisedpublication had to wait for quite a while. Noether was not a quick writer; more oftenher ideas went into the papers of other people rather than forcing herself to write amanuscript. It is not clear which paper she was announcing here; some years laterthere are two publications of Noether on representation theory [Noe29] and [Noe33b](both in the Mathematische Zeitschrift and not in the Mathematische Annalen). Thefirst of these papers [Noe29] consists essentially of the notes taken by van der Waerdenof Noether’s lecture in the winter semester 1927/28. These lecture notes have beensaid to constitute “one of the pillars of modern linear algebra”.53 The second paper[Noe33b] is somewhat more closely related to the topics of her correspondence withBrauer. We have the impression that in the announcement she had in mind one longerpaper but in her hands this became too long and thus was divided into two parts.

Returning to Noether’s letter to Brauer on March 28, 1927: In one of the statementsin that letter she claims that each minimal splitting field of a division algebra D isisomorphic to a maximal commutative subfield of D. But this turned out not to betrue. There do exist minimal splitting fields of D whose degree is larger than theindex; an embedding into D is possible if and only if the degree of the splitting fieldis minimal, i.e., equals the index of D.

We will see that this error had important consequences, leading to the Brauer–Hasse–Noether theorem.

We do not know whether Brauer had replied to Noether’s letter. But we do knowthat both met at the next annual meeting of the DMV, on September 18–24, 1927at the spa of Kissingen. Neither Noether nor Brauer were scheduled for a reportat that meeting but certainly they talked about the topic of Noether’s letter. Fromthe correspondence over the following weeks we can obtain a fairly good picture of

53Cited from [Cur99] who in turn refers to Bourbaki.


what they had discussed in Kissingen. Apparently Brauer knew that Noether’s claimcited above was erroneous, i.e., that there do exist minimal splitting fields whosedegree is larger than the index. And he told Noether. But then Noether asked whetherthe degrees of the minimal splitting fields may be bounded. This too seemed to bedoubtful. Noether wished to check the question at the smallest example, namely theordinary quaternions over the rational number field Q.

In Kissingen they could not settle the question. Two weeks later, on October 4,1927, Noether sent a postcard to Hasse (with whom she had corresponded since1925):

Lieber Herr Hasse, Können Sie mir sagen, ob aus den allgemeinen Exis-tenzsätzen über abelsche Körper direkt dieser folgt: Es gibt zu jedem n

mindestens einen (vermutlich beliebig viele) in bezug auf den Körper derrationalen Zahlen zyklischen Körper des Grades 2n, derart daß sein Un-terkörper vom Grad 2n�1 reell ist, und daß .�1/ in ihm als Summe vonhöchstens drei Quadraten darstellbar ist (Quadrate gebrochener Zahlen).

Dear Mr. Hasse! Can you tell me whether the general existence theoremsfor abelian fields yield the following: For every n there exists at leastone (perhaps arbitrary many) cyclic field over the rational number fieldof degree 2n such that its subfield of degree 2n�1 is real, and it admitsa representation of .�1/ by at most three squares (squares of fractionalnumbers) …54

If such fields would exist then there would exist minimal splitting fields of arbitrarylarge degree. Noether continued:

R. Brauer äußerte (in Kissingen) die Vermutung der Nichtbeschränkt-heit. Seine Beispiele waren aber komplizierter als Quaternionenkörper.Es würde folgen, daß man über diese kleinsten Körper viel weniger weiß,als ich eine Zeitlang dachte.

R. Brauer conjectured in Kissingen the non-boundedness [of the degreesof minimal splitting fields], but his examples were more complicatedthan quaternion fields. It would follow that one knows much less aboutthose minimal fields than I had believed for some time.

Hasse reacted immediately. Already on the next day, on October 6, 1927, he sentto Noether a 4-page manuscript in which he gave a detailed proof that indeed, suchfields do exist. For us it is of interest that his proof was based upon the Local-Global

54Of course, the condition that the subfield of degree 2n�1 should be real, is always satisfied and hence couldbe omitted. This is another instance where we can see that Noether often wrote her postcards very impulsively anddispatched them immediately – without thinking twice about the text. (Very much like some people send e-mailmessages nowadays.) If she had, she would certainly have noticed that the condition to be real is superfluous, asshe admits in her next postcard.


Principle: not the Local-Global Principle for algebras (this was not known yet) butfor quadratic forms, which Hasse had discovered 1922 in his famous habilitationthesis [Has23b]. The quadratic form relevant to Noether’s question is the sum of foursquares: f .x/ D x2

1 C x22 C x2

3 C x23 . The question whether f .x/ has a non-trivial

zero in a field is, of course, identical with the question whether the field is a splittingfield of the ordinary quaternions.

So we see here, in Hasse’s letter of October 1927, the nucleus of what in 1931would become the Local-Global Principle for algebras in the Brauer–Hasse–Noetherpaper.

Immediately thereafter, on October 10, 1927, Noether wrote to Brauer sendinghim Hasse’s letter which solves the question under discussion, and proposing a jointnote, to be published together with a note of Hasse. There followed a series ofletters within the triangle Brauer–Hasse–Noether, discussing details about the plannednotes and possible generalizations. After a while Brauer succeeded to construct thosefields without using Hasse’s Local-Global Principle for quadratic forms. Hasse askedBrauer to explain to him the group theoretic relevance of his example, which Brauerdid in full detail.

Finally there appeared a joint note of Brauer and Noether [BN27], and immediatelyafter it in the same journal a note by Hasse [Has27a]. The offprints of both paperswere bound together and distributed in this form.

Here we see the first instance where Brauer, Hasse and Noether had formed a teamtowards a common goal – as a consequence of Noether’s error concerning minimalsplitting fields.

1.7.2 Hasse’s castles in the air

Thereafter Hasse became increasingly interested in the theory of algebras becausehe had seen that number theory, in particular class field theory and the local p-adictheory, could be used there profitably. Brauer became interested in class field theorybecause of the same reason. And Noether, who had brought the two together in thefirst place, was pleased because she observed that “her methods” were accepted byboth.

There resulted a regular exchange of letters and information between the threemembers of the team. Brauer learned from Hasse about class field theory and Hasselearned from Brauer about algebras and group rings.

For instance, upon a request of Hasse, Brauer wrote him on July 9, 1929 all that heknew about group rings in the setting of algebras. On October 26, 1929 Brauer sentto Hasse his notes which he had composed for his lecture course at Königsberg. OnMarch 16, 1930 Hasse wrote to Brauer explaining in detail his main ideas and resultson skew fields over}-adic fields (they appeared later in [Has31d]). We have discussedthat paper in Section 1.6.1 in connection with the Local Structure Theorem. (See alsoSection 1.6.3.) At the end of his letter Hasse observed that over a local fieldKp (with


p finite) there exist extension fields with non-cyclic Galois group; nevertheless everycentral skew field over Kp is cyclic. And then he asks Brauer:

Ich möchte nun fragen, ob Ihnen über dem rationalen Körper als Zentrumoder einem gewöhnlichen algebraischen Zahlkörper als Zentrum einSchiefkörper bekannt ist, für den es keinen Abelschen oder wenigstenskeinen zyklischen minimalen Zerfällungskörper D maximalen Teilkör-per gibt. Ist etwa das von Ihnen angedeutete direkte Produkt zweierQuaternionenalgebren in diesem Sinne „nicht zylisch“ ?

Now I would like to ask whether you know a central skew field over therational or an algebraic number field which does not admit an abelian, orat least no cyclic, minimal splitting field which is a maximal commutativesubfield. Is it true that the direct product of two quaternion algebras,which you mentioned the other day, is non-cyclic in this sense?

We see that Hasse contemplates about whether globally every central skew field isabelian or perhaps even cyclic, i.e., the Main Theorem. But he is not sure and wantsto know the opinion of Brauer.

In his reply on April 18, 1930 Brauer thanked Hasse for this letter, saying that hewas highly interested in Hasse’s beautiful results on the arithmetic of hypercomplexnumbers. Concerning Hasse’s question he wrote:

Die von Ihnen gestellte Frage kann ich leider zur Zeit noch nicht beant-worten. Ich weiß nicht einmal, ob es Schiefkörper (mit endlichem Rangüber ihrem Zentrum) gibt, die keinen Normalkörper als maximalenTeilkörper besitzen. Ich habe früher vergeblich versucht, die Existenzeines solchen Normalkörpers zu beweisen. Jetzt will ich umgekehrt ver-suchen, ein Beispiel zu konstruieren, bei dem es keinen derartigen Nor-malkörper gibt.

Unfortunately I am not able yet to answer your question. I do not evenknow whether there exist skew fields (of finite rank over their center)which do not admit a normal55 field as a maximal subfield. Formerly Ihave tried without success to prove the existence of such a normal field.Now I am trying the opposite, to construct an example which does notadmit such a normal field.

But, he adds, although he has an idea how to construct such an example, this willprobably be very sophisticated. Apparently he did not succeed. To our knowledge,such an example was first given by Amitsur [Ami72]. But, of course, the base fieldin Amitsur’s construction was not an algebraic number field.56 Brauer continues:

55A “normal” field extension, in the terminology of the time, means a Galois extension.56Since that time there has developed an extensive literature trying to understand the construction of non-


Auch die andere von Ihnen gestellte Frage, ob es Schiefkörper gibt, dienicht vom Dicksonschen Typ sind (d.h. die keinen zyklischen maximalenTeilkörper haben), weiß ich nicht. Das von mir erwähnte Produkt derbeiden Quaternionensysteme scheidet aus, da das Zentrum dabei nichtalgebraisch ist … Sobald ich genaueres weiß, schreibe ich Ihnen dannnoch einmal.

Also your more specific question whether there are skew fields whichare not of Dickson type (i.e., which have no cyclic maximal subfields),I am not able to answer. The product of two quaternion algebras whichyou mentioned, is not eligible since its center is not algebraic…As soonas I will know more about it I will write again to you.

The said “product of two quaternion algebras” had been treated by Brauer in his paper[Bra30b] over the rational function field Q.u; v/ of two variables. The examplesprovided by Brauer have exponent 2 and index 4 but the question whether they arecyclic is not treated in [Bra30b]. The first who explicitly constructed a non-cyclicdivision algebra was Albert [Alb32a].57

At the end of his letter Hasse said that he would very much like to talk personallyto Brauer about these questions. And he announced that in the fall he will be inKönigsberg (the place where Brauer lived) for the meeting of the DMV (GermanMathematical Society). We may safely assume that Hasse and Brauer had a verythorough discussion there, together with Emmy Noether who also participated at themeeting. Unfortunately there is no record about their conversations.58

In December 1930 Hasse seemed to have made up his mind and written up somecoherent conjectures about the structure of algebras over number fields. He did so ina letter to Emmy Noether. We do not know Hasse’s letter but we do know the reactionof Emmy Noether. From that we can conclude that among Hasse’s conjectures wasthe Main Theorem and the Local-Global Principle, but also the consequence that overnumber fields, the index of a central simple algebra equals its exponent. It appearsthat Hasse had mentioned that his conjectures do not yet have a solid foundation. ForNoether replied on December 19, 1930 :

Ja, es ist jammerschade, daß all Ihre schönen Vermutungen nur in derLuft schweben und nicht mit festen Füßen auf der Erde stehen: denn ein

crossed product division algebras. As a noteworthy example we mention the work by Brussel who uses Wang’scounterexamples to Grunwald’s theorem to construct non-crossed products. See [Bru97], and also subsequentpapers of the same author.

57Albert used a construction similar to but not identical with Brauer’s. Note that in a footnote on the first pageof [Alb32a] it is claimed that Brauer’s construction was false, but at the end the author admitted in an additionalnote that the difficulty was one of the interpretation of language, rather than a mathematical error.

58Except Noether’s reference to their trip to Nidden of which she reminded Hasse in her letter of November 12,1931 which we have cited in Section 1.4.3. – As a side remark we would like to draw the reader’s attention tothe essay [Tau81] by Olga Taussky-Todd who reported about her experiences during this Königsberg meeting,including her vivid recollection of how Noether and Hasse seemed to have a good time discussing her (Olga’s)results on the capitulation problem of class field theory.


Teil von ihnen – wieviel übersehe ich nicht – stürzt rettungslos ab durchGegenbeispiele in einer ganz neuen amerikanischen Arbeit: Transactionsof the Amer. Math. Society, Bd. 32; von Albert. Daraus folgt zunächst,daß der Exponent wirklich kleiner sein kann als der Index, schon beimrationalen Zahlkörper als Zentrum; und damit also weiter daß Ihre For-mentheorie sich nicht auf Formen höheren Grades übertragen läßt. ObIhre Vermutung mit dem zyklischen Zerfällungskörper gilt, wird zummindesten zweifelhaft.

Yes, it is a terrible pity that all your beautiful conjectures are floating inthe air and are not solidly fixed on the ground: for part of them – howmany I do not yet see – hopelessly crash through counterexamples ina very new American paper … by Albert. From that it follows, firstly,that the exponent can indeed be smaller than the index, already withthe rational number field as center, and furthermore that your theory offorms cannot be transferred to forms of higher degree. Whether yourconjecture concerning cyclic splitting fields holds is at least doubtful.

After this, Noether proceeds to explain to Hasse the counterexamples which shepurports to have found in Albert’s paper [Alb30]. And she ends the letter by askingHasse to inform her if his cyclic splitting field is crashed too; this is an indicationthat indeed the Main Theorem was conjectured by Hasse, as we had stated abovealready.

We see that Noether did not yet believe in the validity of the Main Theorem ofwhich one year later she would be a co-author.

In her letter Noether had mentioned Hasse’s “theory of forms” (“Formentheorie”).We do not know precisely what was meant by this. In the present context it seemsprobable that “forms” were to be understood as norm forms from central simplealgebras, and that Hasse had the idea that some kind of Local-Global Principle shouldhold for those norm forms – in analogy to quadratic forms which he had treated in histhesis and following papers. If our interpretation is correct then we can conclude thathere, in December 1930, Hasse’s conjectures included the Local-Global Principle foralgebras.

The vivid language Noether had used in her reply appears to be quite typical ofher style. Hasse apparently did not mind it. He seems to have checked Albert’s papercited by Noether and found out that Noether’s interpretation of Albert’s result wasincorrect. And he wrote this to her. Whereupon on December 24, 1930 (Christmaseve!) she replied:

… im übrigen aber ist dieser Brief ein pater peccavi. Ihre Luftschlössersind nämlich noch garnicht umgefallen … ich habe aus der Albertschen„Fürchterlichkeit“ ziemlich genau das Gegenteil dessen herausgelesenwas drinstand. Erst Ihr Gegenbeispiel hat mir die Sache klar gemacht.


This letter is a pater peccavi. For, your castles in the air are not yetcrashed … I have extracted from Albert’s horrible thing exactly the op-posite of what was in it. 59 Only your counterexample has cleared up thesituation for me.

Noether said that she had extracted from Albert’s paper the opposite of what was init. After longish explanations of her error she writes:

Es scheint also doch ganz wahrscheinlich, daß bei algebraischen Zahlkör-pern als Zentrum immer Exponent und Index übereinstimmen.

After all, it appears probable that with an algebraic number field ascenter, exponent and index do always coincide.

From then on, Noether was on Hasse’s side and she vividly advocated his conjec-tures. She arranged that Hasse was invited to give a colloquium talk in the GöttingenMathematical Society on January 13, 1931. The title of his talk was “On skew fields”(“Über Schiefkörper”). Hasse’s manuscript for this talk is preserved. This shows:

In Göttingen, on January 13, 1931 Hasse publicly announced his con-jecture for the Main Theorem.60

Although Hasse in his manuscript did not say anything about how he would try toapproach this conjecture, implicitly we can see the Local-Global Principle behindit. For, he reports extensively on his results about local algebras [Has31d], and thathe had discovered they are always cyclic. And starting from the local results heformulates the conjectures for the global case.

1.7.3 The Marburg skew congress

Some weeks later Hasse and Noether met again, this time in Marburg on the occasionof a small congress, today we would say workshop, on skew fields.61 Hasse’s ideawas to bring together mathematicians who were active in the theory of algebras, ofclass fields and of group representations, in order to join forces with the aim of solvinghis conjectures. It seems that he had discussed this plan with Emmy Noether whenhe visited Göttingen for his colloquium talk in January; they had agreed to have theworkshop at the end of the winter semester, February 26 to March 1, 1931. Noetherliked to call the meeting the “skew congress”.

59Albert’s style of writing was mostly using extended computations which Noether hated; she was advocatingthe use of abstract notions and structural arguments. This explains that she called his paper “horrible”.

60In the book [Cur99] it is said that the Main Theorem had been conjectured by Dickson already. But we didnot find this in Dickson’s works, and after inquiring with the author he replied: “The statement in my book aboutDickson’s conjecture has to be withdrawn.” His statement was based on an assertion in Feit’s obituary article onBrauer [Fei79] which he had accepted without checking.

61Starting with the summer semester 1930 Hasse had accepted a professorship in Marburg as the successor ofKurt Hensel. On this occasion he had been granted some funds for inviting visitors to Marburg for colloquiumlectures etc.


Of course Richard Brauer was invited too. In the letter of invitation to Brauer,dated February 3, 1931, Hasse said this will become a small, “gemütlicher” congresson skew fields.62 Hasse mentions the names of the other people who were invited:Noether, Deuring, Koethe, Brandt and Archibald (disciple of Dickson). Moreover,Hasse wrote, invitations will be sent to all people who are interested, e.g., Artin,Speiser, I. Schur – but the funds available to Hasse were not sufficient to cover theexpenses of all of them. From the correspondence of Hasse with Krull we knowthat Krull was invited too, as well as F. K. Schmidt (both in Erlangen) but they wereunable to come because at precisely the same day they had invited a guest speaker tothe Erlangen colloquium.

Emmy Noether, in her letter to Hasse dated February 8, 1931 offered a title forher own talk, and she forwarded already some proposals for the program of the skewcongress. She proposed the talks of R. Brauer, Noether, Deuring, Hasse to be held inthis order, so that every one of the lecturers could build on the foregoing talks. Theother lectures, she wrote, were independent. In her letter she also mentioned Fitting,a Ph.D. student of hers.63

Moreover, Noether strongly recommended to invite Jacques Herbrand, her Rocke-feller fellow. She wrote that Herbrand had worked on Logic and Number Theory only.Number Theory he had learned from Hasse’s Class Field Report and Hasse’s paperson norm residues. She recommends him as a mere participant only but if he is to givea talk too then he could report on his results about the integral representation of theGalois group in the group of units [Her30].

In those years it was not uncommon that colloquium lectures, meetings etc. ofthe foregoing year were reported in the Jahresbericht der DMV. Accordingly, in its1932 issue we can find the Marburg 1931 skew congress program as follows (Englishtranslation):64

62The following passage from Hasse’s letter to Brauer may also be of interest: “As I told you already lastsummer, we are doing representation theory this year, and we use mainly your lecture notes which you had kindlysent me.”

63Recently some letters from Emmy Noether to Paul Alexandrov have been published by Renate Tobies[Tob03], and there, in a letter of October 13, 1929, we read: “Hasse will go to Marburg as the successor ofHensel; I wrote to him concerning connected visiting lectures but have not yet obtained a reply.” In her commentsTobies interpreted this as Noether having written to Hasse proposing to establish in Marburg something like the“skew congress”. But this interpretation remains doubtful. Among the letters from Noether to Hasse of that timewe did not find any passage of that kind. Instead, a few days earlier than her letter to Alexandrov she wrote aletter to Hasse (October 7, 1929) where we read something else. There she first congratulates Hasse for havingreceived the offer from Marburg University, and then she proposed Alexandrov to be named as Hasse’s successorin Halle. It is not clear how this blends with what she wrote to Alexandrov about “connected visiting lectures”(“zusammenhängende Gastvorlesungen”). We could speculate that she hoped, if Alexandrov would have beenmentioned in the list of possible successors of Hasse in Halle, then at least she could obtain funds for invitinghim to Germany for longer periods. But this is pure speculation and so the meaning of Noether’s writing toAlexandrov remains in the dark, as for our present knowledge. – In any case, as we see, once Noether knewabout Hasse’s idea for this skew congress, she actively stepped in and helped him in the planning.

64We have found the reference to this program in Tobies’ article [Tob03].


26.2. H. Hasse Dickson skew fields of prime degree.

27.2. R. Brauer (Königsberg) Galois theory of skew fields.

M. Deuring (Göttingen) Application of non-commutative algebra to norms andnorm residues.

E. Noether (Göttingen) Hypercomplex structure theorems and number theo-retic applications.

R.Archibald (Chicago) The associativity conditions in Dickson’s division al-gebras.

H. Fitting (Göttingen) Hypercomplex numbers as automorphism rings ofabelian groups.

28.2. H. Brandt (Halle) Ideal classes in the hypercomplex realm.

G. Köthe (Münster) Skew fields of infinite degree over the center.

There may have been other participants in the workshop who did not deliver atalk. Perhaps Herbrand was one of them.

We see that Noether’s proposals for the order of the talks were not realized;perhaps the mutual dependance of the talks was not so strong that a unique orderwould follow. The first day (February 26) was the day of arrival; we know that thevisitors arrived late at noon and hence Hasse’s talk had probably been scheduledsome time in the afternoon. We observe that in the title he used the old terminology“Dickson algebras”. Hasse’s notes for this lecture are preserved and there, however,he speaks of “cyclic algebras”. From the notes we infer that Hasse presented areport on cyclic algebras of prime degree, based on his earlier work [Has31d] and theHilbert–Furtwängler Norm Theorem (see Section 1.4.1). At the end he presented anumber of problems, thus setting the pace for this conference and for future work inthe direction of the Local-Global Principle (see Section 1.4) and the structure of theBrauer group (Section 1.6).

If Hasse’s aim in this workshop had been to get a proof of the Main Theorem thenthis aim was not achieved. But there resulted a general feeling that the final solutionwas near. Hasse, in particular, seems to have been encouraged and kept himself busyworking on the problem.

Already a week later, on March 6, 1931 Hasse proudly sent a circular to theparticipants of the skew congress with the following telegram style message:

Liebe(r ) Herr/Fräulein: Habe soeben den fraglichen Normensatz fürzyklische Relativkörper bewiesen, und mehr braucht man für die Theo-rie der zyklischen Divisionsalgebren nicht.

Dear Sir/Madam: Just now I have proved the norm theorem in questionfor relatively cyclic fields, and more is not needed for the theory of cyclicdivision algebras.


In other words: He had proved the Local-Global Principle to hold for cyclic algebrasof any index, not necessarily prime. This was one of the problems which he hadstated in his talk on February 26. Hasse published the result in [Has31a], and itsconsequences for cyclic algebras were announced in [Has31c]. A complete theory ofcyclic algebras over number fields followed in his American paper [Has32c].

From here on, we have already reported the further development in the foregoingSections 1.4–1.6.

1.8 The American connection: Albert

1.8.1 The footnote

There is an extended footnote in the Brauer–Hasse–Noether paper [BHN32] whichreads as follows (in English translation):

The idea of reduction to solvable splitting fields with the help of Sylow’sgroup theoretical theorem has been applied earlier already by R. Brauer,namely to show that every prime divisor of the index also appears in theexponent [Bra28]. Recently A.A.Albert has developed simple proofsfor this idea, not dependent on representation theory, also for a numberof general theorems of the theory of R. Brauer and E. Noether ([Alb31c],[Alb31b]; for the reduction in question see in particular Theorem 23 in[Alb31b]).

Added in proof. Moreover A.A.Albert, after having received the newsfrom H. Hasse that the Main Theorem has been proved by him for abelianalgebras (see in the text below), has deduced from this, independent fromus, the following facts:

a) the Main Theorem for degrees of the form 2e ,b) the theorem 1 below (exponent = index),c) the basic idea of reduction 2, and also of the following reduction 3,

naturally without referring to reduction 1, and accordingly with theresult: for division algebras D of prime power degree pe over �there exists an extension field �0 of degree prime to p over �, sothat D�0 is cyclic.

Of course, all three results are now superseded by our proof of themain theorem which we have obtained in the meantime. But they showthat A.A.Albert has had an independent share in the proof of the maintheorem.


Finally, A.A.Albert has remarked (after knowing our proof of the maintheorem) that our central theorem I follows in a few lines from thetheorems 13, 10, 9 of a paper which is currently printed ([Alb31d]). Theproof of those theorems is based essentially on the same arguments asour reductions 2 and 3.

Here, the “reductions” are the steps in the proof of the Main Theorem in the Brauer–Hasse–Noether paper. As explained in Section 1.4.2, “reduction 2” is Brauer’s Sylowargument, “reduction 3” is Noether’s induction argument in the solvable case.

This footnote has aroused our curiosity. We wanted to know more about therelation of Albert to Hasse, and about Albert’s role in the proof of the Main Theorem.The correspondence between Albert and Hasse during those years is preserved. Ourfollowing report is largely based on these documents.

To have a name, we shall refer to this footnote as the “Albert-footnote”. For laterreference we have divided the Albert-footnote by horizontal lines into three parts. Itwill turn out that these parts were added one at a time. The horizontal lines are notcontained in the original. Also, in the interest of the reader we have changed thereferences to Albert’s papers in the original footnote to the corresponding referencecodes for this article.

1.8.2 The first contacts

A.Adrian Albert (1905–1972) had been a disciple of Dickson. We have said inSection 1.6.3 (p. 40) already that Dickson’s book Algebras and their Arithmetics hada great influence on the work of German mathematicians in the 1920s. The resultsof Dickson and his disciples were noted carefully and with interest by the Germanmathematicians around Noether.

Already in Section 1.7 (p. 51) we have met the name of Albert. There we reportedthat Noether, after reading a recent paper by Albert, thought erroneously for a shorttime that, with the help of Albert’s results, she could construct a counterexample tosome of Hasse’s conjectures in connection with the Main Theorem.

We find the name of Albert again in the manuscript which Hasse had written forhis personal use on the occasion of the colloquium talk at the Göttingen MathematicalSociety, January 13, 1931. This was the talk where Hasse publicly announced forthe first time his conjecture for the Main Theorem; we have discussed it on p. 53.After having announced his conjecture, Hasse, according to his manuscript, reportedwhat was known for division algebras D of small index n. For n D 2 and n D 3

he mentioned Wedderburn and Dickson, respectively, for the fact that D is cyclic.For n D 4 he cited Albert for the fact that every division algebra contains a maximalsubfield of degree 4 which is abelian with Galois group non-cyclic of type .2; 2/.But Hasse noted in his manuscript that Albert’s proof cannot be valid because “thereexist, as is easily seen, cyclic division algebras of index 4 which do not contain anabelian subfield with group of type .2; 2/.”


We do not know what Hasse had in mind when he wrote “as is easily seen”. Hisown Local-Global Principle, which he conjectured at this colloquium, can be usedto prove that his assertion is not true over number fields. In any case, we observethat Hasse did not write that Albert’s proof “is not valid” but he wrote it “cannot bevalid”; this indicates that he had not checked Albert’s proof in detail but had in mindsome construction of those algebras which would yield a counterexample to Albert’sassertion. We do not know when Hasse had discovered that his construction did notwork. Maybe this was shortly before he actually went to Göttingen, and then clearlyhe would not have mentioned it in his talk. Maybe it was after his talk in the discussionwith Emmy Noether who, having had her own experience with Albert’s paper (as wehave seen in the foregoing section), had now studied it in detail and could assureHasse that it was correct. In any case we know that Hasse, after this experience, nowwished to establish contact with Albert in order to clear up the situation.

We do not know Hasse’s first letter to Albert, or the precise date when it was sent.Albert’s reply is dated February 6, 1931 and so, taking into consideration the timefor overseas postal delivery65 Hasse wrote his letter shortly before or shortly afterhis Göttingen colloquium lecture on January 13. Thus started the correspondencebetween Hasse and the 7 years younger Albert, which continued until 1935. Thereare 15 letters preserved from Albert to Hasse and 2 letters in the other direction.

As we learn from Albert’s reply, Hasse had addressed his first letter to Dickson(with whom he had exchanged reprints in the years before) who forwarded it toAlbert.At that time Albert was at Columbia University in New York. Hasse had describedhis own work in his letter; Albert replied that he was very interested in it and heintroduced himself. The next letter of Albert is dated March 23, 1931. As we havesaid earlier, only the letters from Albert to Hasse are preserved while most of theletters from Hasse to Albert seem to be lost. Accordingly, when in the following wereport about letters from Albert to Hasse we have to remember that usually betweentwo such letters there was at least one from Hasse toAlbert. 66 Now, on March 23 thereis already some mathematical discussion in the letter. Replying to Hasse’s questionon the existence of non-cyclic division algebras of index 4 over a number field, Albertwrote:67

The question seems to be a number-theoretic one and I see no way to getan algebraic hold on it. It seems to be a hopeless problem to me aftermore than a year’s work on it.

We observe that this is the same question which Hasse had asked Brauer a year before(letter of March 16, 1930) but could not get an answer either (see p. 50). It appears thatthe motivation of Hasse was to secure his conjecture concerning the Main Theorem.

65This was about two weeks – there was no air mail yet.66Sometimes there were more than one, for Albert on June 22, 1932 wrote: “You are, may I say it, a very

pleasing friend to write me so often without receiving any answer.”67The letters between Albert and Hasse were written in English. We are citing directly from the original.


If the experts were not able to construct non-cyclic algebras over a number field thenthis would add some point to the conjecture being true.

Albert also reported on the results of his new paper [Alb31c] which had appearedin the January issue of the Transactions of the AMS. Every central division algebra ofindex 6 68 over any field of characteristic 0 is cyclic – provided it satisfies what he callsa mild assumption R2 (and which he could remove in his later paper [Alb31b]). Inreply to some other related question, namely about the product of two central simplealgebras, Albert presented an erroneous answer, saying that A˝K A is a total matrixalgebra. But he corrected this four days later, writing that it had to be the product ofA with the reciprocal algebra A0. 69

This and more was of course known already to the trio Brauer–Hasse–Noether,by Brauer’s theorems of 1928 which we have cited in Section 1.4.2 (p. 13), andalso by Emmy Noether’s Göttingen lectures 1929/30 – which however, were not yetpublished.70 In the next letter, dated May 11, 1931,Albert wrote that he had completeda paper containing most of Brauer’s results but which he had obtained independentlyand with his own new methods. Fortunately, he added, he had discovered Brauer’spapers before it was too late and hence could give Brauer priority. This refers toAlbert’s paper [Alb31b]. The methods of Albert in this paper are independent ofrepresentation theory and, in this sense, they can be regarded as a simplification ofBrauer’s approach.

We see that Hasse’s initiative to open direct contact between German andAmericanmathematicians working on algebras, had from the start been accepted by Albert. Wehave to be aware of the fact that at the time there were not many international meetingsto establish contacts, no e-mail, and journals arrived usually much later than the timewhen the results had been discovered. The letters of Albert show that he was fullyaware of the possibilities which the correspondence with Hasse opened to him: Topresent his ideas and results to Hasse (and hence to the German group working onalgebras) and at the same time to learn about their methods and results (which heregularly shared with Dickson and Wedderburn). In his letter of May 11 he wrote:

Your work on quadratic forms is not new to me. In fact I have beenreading your Crelle and Jahresbericht work ever since your first letter tome. In this period I have also been able to apply your most fundamentalresult on quadratic forms in n � 5 variables, together with my abovementioned new methods to prove the following results …

It appears that Hasse had explained to Albert his idea of using the Local-GlobalPrinciple for quadratic forms, perhaps in a similar way as he had written to Brauer

68Albert spoke of algebras of “order 36”, thereby defining the “order” as the dimension of the algebra as vectorspace over the base field. We shall avoid this terminology which is in conflict with the terminology used byNoether and Hasse.

69Albert corrected this error in the already published paper [Alb31c] by putting a page of Errata into the sameTransactions volume 33.

70See [Noe29], [Noe33b], [Noe83], [vdW31].


(see p. 19). And Albert had reacted immediately, proving theorems about quaternionalgebras and algebras of 2-power index which are the obvious candidates whose normforms may possibly be handled by quadratic forms. Among the results which Albertreports in this letter is his answer to Hasse’s former question which he (Albert) hadclassified as hopeless even in his foregoing letter, namely:

Every central division algebra of index 4 over an algebraic number fieldis cyclic.71

Moreover he writes that, over a number field, the product of two quaternion algebrasis never a division algebra, and that the same is true for division algebras of 2-powerindex. Finally, for division algebras of 2-power index he proved Hasse’s exponent-index conjecture.

Albert adds that the last of the above statements would probably remain true if2 is replaced by any prime number p provided he could prove some kind of Local-Global Principle for the norm forms of division algebras of index p. But in the nextletter (June 30) he apologizes for having given the impression that he could prove theexponent-index conjecture generally. He is still working on it. Also he writes:

I want to remark in this connection that I have proved that your resultsimply that the direct product of any two central division algebras is adivision algebra if and only if the indices of the two algebras are relativelyprime (for an algebraic reference field).

And so on. The letters from Albert are full of information about his results, some ofthem obviously following Hasse’s suggestions. Albert writes:

The work of the German mathematicians on algebras is very interestingto me and I should like to know all of it if possible … and am very pleasedand thankful for the opportunity to communicate with you and know ofyour results.

In this connection we have to mention Hasse’s “American paper” [Has32c]. Wehave already had several occasions to cite this paper. It contains a comprehensivetreatment of cyclic algebras over number fields. The Local-Global Principle for cyclicalgebras over number fields is cited there and used in an essential way. We have seenin Section 1.7.3 (p. 55) that it was around March 6, 1931 when Hasse, as a follow-upto his skew congress, had obtained the Local-Global Principle for cyclic algebrasof arbitrary index. On the other hand, Hasse’s American paper was received by the

71One year later, in April 1932 Albert presented to the American Mathematical Society for publication aconstruction of non-cyclic division algebras of index 4, defined over the rational function of two variables[Alb32a]. We have already mentioned this in the foregoing section; see footnote 57). These algebras haveexponent 2. Some months later, in June 1932, he was able to give a refined construction. this time of non-cyclicdivision algebras which have both index and exponent 4. Albert was 26 then, and throughout his career heseemed to have been quite proud of this achievement.


editors on May 29, 1931. Thus Hasse had conceived and completed this paper inabout two months. In an introductory paragraph to this paper Hasse says:

I present this paper for publication to anAmerican journal and in Englishfor the following reason:

The theory of linear algebras has been greatly extended through thework of American mathematicians. Of late, German mathematicianshave become active in this theory. In particular, they have succeededin obtaining some apparently remarkable results by using the theoryof algebraic numbers, ideals, and abstract algebra, highly developedin Germany in recent decades. These results do not seem to be as wellknown inAmerica as they should be on account of their importance. Thisfact is due, perhaps, to the language difference or to the unavailabilityof the widely scattered sources …72

Reading this text and knowing that it has been written in the months March toMay 1931 when the first letters Hasse-Albert were exchanged, we cannot help feelingthat to write this paper in English and to publish it in an American journal, was meantpredominantly as a source of information for Hasse’s correspondence partner Albert.It seems that Hasse had observed Albert’s high qualifications and great power as amathematician, and he knew that Albert was eager to absorb the methods and resultswhich had been developed in Germany. And so when Hasse wrote the paper, he hadin mind Albert as the first and foremost reader. In fact, Albert informed him (letterof November 6, 1931) that he “was fortunate to read your paper for the editors of theTransactions”. In other words: Albert had to referee the paper, and so he was the firstin America to know its contents, long before the paper finally appeared in 1932.73

Between Albert’s letter of June 30, 1931 and his next letter of November 6 there isa gap of several months. On Hasse’s side, this gap can perhaps be explained becausein the summer semester of 1931 he had Harold Davenport as a visitor from England.Hasse had invited Davenport to stay in his Marburg family home under the conditionthat they would speak English only, so Hasse could refresh and upgrade his knowledgeof the English language. It is conceivable that Hasse, besides his mathematicalactivities, was now absorbed in his English studies.74 Moreover, as is well known,the beginning friendship between Hasse and Davenport induced Hasse to become

72The last mentioned fact seems to have been quite serious in those times. Even a big and renowned institutionlike Columbia University of New York (where Albert stayed in the summer of 1931) did not have the journalHamburger Abhandlungen with the important papers of Artin in its library; for in his letter of May 11, 1931Albert asks Hasse for information about what is contained in those papers because he was not able to obtainthem.

73In [Cur99] it is said (p. 232) that the results mentioned in Hasse’s introductory paragraph had already beenwell known toAlbert. We believe that theAlbert–Hasse correspondence of 1931 shows that the Hasse letters werewhat stimulated Albert to study eagerly in more detail the work on algebras which was conducted in Germany.

74Perhaps Hasse had written to Albert about his English studies, for in the letter of June 30 Albert wrote: “YourEnglish is very clear and understandable. I only wish I could write German half so well!”


interested in Davenport’s work on the solution number of congruences, which laterled to Hasse’s proof of the analogue of the Riemann hypothesis for elliptic functionfields – thereby starting a series of seminal papers of Hasse on algebraic functionfields. We know that after the end of the summer semester 1931 the Hasses togetherwith Davenport went on an extended tour through central Europe (in Davenport’s car)which ended only in September at the DMV-meeting in Bad Elster.75 In view of thisit is plausible that in this summer Hasse did not find the time to work intensively onalgebras and to write letters about it.

However in October 1931 Hasse seemed to have again taken up his work onalgebras. And he informed Albert about his results. Albert’s reply of November 6starts with the following text:

I received your very interesting communication this morning and wasvery glad to read of such an important result. I consider it as certainlythe most important theorem yet obtained for the problem of determiningall central division algebras over an algebraic number field.

What was the result that Hasse had communicated to him, which Albert classified as“the most important result yet obtained … ”? Taking into account the delivery timefor overseas mail, we conclude that Hasse may have dispatched his letter aroundOctober 20. At that time, as we know, Hasse had not yet obtained a proof of the MainTheorem. But we remember from Section 1.3 (p. 7) that Hasse had already succeededwith the proof that every abelian algebra is cyclic, and that he had informed EmmyNoether about it. He had also informed Brauer. Now we see that Hasse had alsoinformed Albert at the same time.

We conclude that by now Hasse had accepted Albert as a correspondence partneron the same level as he had Richard Brauer and Emmy Noether. Thus the “triangle” ofBrauer, Hasse and Noether had become a “quadrangle” by the addition of Albert, thelatter however being somewhat apart because of the longer distance, which implieda longer time for the transmission of mutual information.

This disadvantage of longer distance became apparent soon.

1.8.3 Albert’s contributions

Albert in his above mentioned letter of November 6 informed Hasse about his resultswhich he had obtained during the summer (when there was no exchange of letterswith Hasse), some of which he had already submitted for publication. Together withHasse’s new results on abelian algebras they would lead to interesting consequences,Albert wrote. And he proposed a joint paper with Hasse.

But before this letter reached its destination, Hasse had found the proof of thefull Main Theorem. We know from Section 1.3 that this happened on November 9,

75For all this see our paper [Roq04].


when Hasse had received from Brauer and Noether the relevant information. The newproof 76 turned out to be so simple (“trivial” as Noether had called it) that Hasse’sformer results and methods developed in this direction became obsolete. ThusAlbert’sletter of November 6 was now superseded by the new development. Nevertheless,Albert may be conceded an “independent share” in the proof of the Main Theorem,as was expressed in the Albert-footnote. Let us describe this in some more detail.

The Albert-footnote consists of three parts. The first part was written when Hassecomposed the first draft for the Brauer–Hasse–Noether paper; this was on Novem-ber 10 as we have seen in Section 1.3 (p. 9). At this time Hasse had not yet receivedAlbert’s letter of November 6, and so he mentioned Albert’s contributions whichhe knew at that time, i.e., those which were contained in the Transactions papers[Alb31b] and [Alb31d]. These papers had been announced to him by Albert in hisletters of March 23 and May 11 respectively. In particular, “theorem 23” of [Alb31d]is mentioned in the Albert-fotnote. This theorem reads as follows:

Theorem 23. Let A be a central division algebra over K of primepower index ps > 1, and M a maximal commutative subfield of A.Then there exists a field extension L0jK of degree prime to p such thatAL0

D A ˝K L0 is a central division algebra over L0 with maximalcommutative subfieldL D M ˝K L0, such that there is a chain of fields

L0 � L1 � � � � � Ls�1 � Ls D L

where each Li jLi�1 is cyclic of degree p (1 � i � s).77

Comparing this with the reductions of Brauer and Noether as presented in section 1.4.2(p. 14) we observe that both statements are very similar to each other. Moreover,Albert’s proof of “theorem 23” contained the same ingredients as the Brauer–Noether“reductions 2 and 3”, namely a Sylow argument (like Brauer) and some kind ofinduction argument from L0 to L (like Noether). This was the reason why Hasse inthe Albert-footnote mentioned “theorem 23” in connection with Brauer and Noether.

But at that time it seemed not yet to be clear whether “theorem 23” indeed wassufficient to replace completely the Brauer–Noether arguments. This question wascleared up only later when Albert’s letter of November 6 had reached Hasse.

Before discussing this and the later letters of Albert let us report on the reaction ofEmmy Noether when she read Hasse’s Albert-footnote. We recall from Section 1.3that Hasse had sent a draft of their joint paper to Noether, and she commented on it inher replies. In her letter of November 12 she writes concerning the Albert-footnote(i.e., its first part):

76More precisely: that part of the proof which consisted in the reductions 2 (Brauer) and 3 (Noether).77The notation is ours, not Albert’s. – Albert formulates this theorem for base fields K of characteristic 0

only, but from the proof it is clear that it holds for any field of characteristic ¤ p.


Das müssen Sie aber bei Albert abschwächen; er hat, Satz 19, nur fürden Fall zyklischer Algebren gezeigt daß jeder Primteiler des Index imExponenten aufgeht; von der allgemeinen Brauerschen Reduktion kannich wenigstens nichts finden.

But you have to weaken your reference to Albert; he has (theorem 19)shown for cyclic algebras only that every prime divisor of the indexdivides the exponent, and I cannot find anything of the general Brauerreduction.

Theorem 19 refers to the paper [Alb31c] of Albert. It seems that Hasse in his replyprotested and pointed out that Albert indeed had essentially the full theorem in ques-tion. For in Noether’s next letter on November 14, obviously replying to Hasse’s“protest”, she writes:

Ich habe Albert noch einmal eingesehen: auch in Satz 20 handelt es sichnur um zyklische Algebren; … Auch später bleibt die Voraussetzung desZyklischen …

I have again checked Albert; also in Theorem 20 only cyclic algebras areinvestigated; …And later too he keeps the assumption of cyclicity …

And she proposes that Hasse should change the footnote; it should be said that Albertdid not have the full result, only in the cyclic case. But Hasse seems to have insistedon his point of view, and to have explained the situation to Emmy Noether. For inher next letter of November 22, she wrote:

Gut daß Sie die Sache mit Albert in Ordnung gebracht haben: da die Heftenoch ungebunden waren, dachte ich nicht daran die übrigen einzusehen,als ich die Arbeit zu haben glaubte. Er scheint mir also wirklich etwaszu können! Mit der Fußnote bin ich ganz einverstanden.

It is good that you have settled the Albert case. Since the fascicles werestill unbounded it did not come to my mind to look at the others since Ibelieved I had the paper in question. It now seems to me that he reallyis very able!78 Now I quite agree with your footnote.

In other words: The volume of the journal in question (Transactions AMS, vol. 33)came in several parts (the whole volume had 999 pages!) and those parts were stillnot bound together in the Göttingen library when Noether looked for Albert’s paper.Noether had studied only one of those parts and, hence, read only one of Albert’spapers.79 So she was not aware about all the relevant results of Albert. She had beenadvised of this by Hasse and now she was happy that Hasse had settled the case inthe footnote. And Noether added the remark:

78Certainly, this comment from Emmy Noether means high praise for Albert.79Besides the two papers [Alb31b], [Alb31c] which are of relevance here, Albert had a third paper in the same

volume, namely [Alb31d].


Daß übrigens alle Leute den Beweis finden, kommt einfach daher, daßSie ihn gefunden haben. Denn das noch Fehlende war trivial für jeden,der nicht wie Sie in die Sache verbohrt war.

By the way, the fact that now all people find this proof, is a consequenceof the fact that you have found it first. What was lacking was trivial foreverybody who was not completely absorbed in the details of proof asyou have been.

Here she refers to the fact that in the case of cyclic algebras Hasse had already provedthe Local-Global Principle in [Has32c]. And the generalization to arbitrary algebrasshe now considered as being “trivial for everybody” (although she and Brauer andAlbert had had a hard time to do it). From today’s viewpoint we would agree with her.But we have already stated earlier that we should not underestimate the difficultieswhich former generations of mathematicians had to overcome before they could settlethe questions which seem to be trivial for us today.

Comparing the dates: The last mentioned Noether letter had been written onNovember 22, in reply to a letter of Hasse. Albert had dispatched his letter to Hasseon November 6. So we may assume that Hasse had received Albert’s letter aroundNovember 20, upon which he had inserted the second part (“Added in proof”) intothe Albert-footnote, and had Noether informed about this. Noether’s letter of Novem-ber 22 which we have cited above would have been her reply to this.

In this second part of the Albert-footnote Hasse listed the three results a), b),c) of Albert, which Albert had mentioned to him in his letter of November 6 (seepage 56). However there arises some question concerning statement c). We havechecked Albert’s letter and found that Hasse’s statement is precisely as Albert hadwritten to him. But this statement is not what Albert has proved in his papers andwhat in later letters he referred to. The difference is that in statement c) the degreeof � 0 (in Hasse’s notation) over the base field is required to be prime to p, whereasAlbert later in his letters and in his work80 does not insist that this is the case. In fact,using the above “theorem 23” it is not difficult to show that the field� 0 WD Ls�1 hasthe property that D ˝K Ls�1 is similar to a cyclic division algebra of index p. Butthe degree ŒLs�1 W K� is divisible by ps�1 and not prime to p.

This weak form of c) (where it is not required that the degree of� 0 is prime to p)appears as theorem 13 in Albert’s [Alb31a], so let us simply call it “theorem 13” inthe following, as Albert does in his letters. This “theorem 13” is quite sufficient forthe proof of the Local-Global Principle in case K is an algebraic number field, as iseasily seen.

For, suppose that the non-trivial division algebraD overK splits everywhere; wemay suppose without restriction of generality that the index of A is a prime powerps > 1. According to “theorem 13” there is a finite field extensionK 0 ofK such that

80Including Albert’s Colloquium Publication Structure of algebras [Alb39].


D0 D D ˝K K 0 is of index p and has a cyclic splitting field. Now, since D splitseverywhere so does D0. By the Hilbert–Furtwängler Norm Theorem it follows thatD0 splits, i.e. has index 1. Contradiction.

We see that indeed, the Local-Global Principle follows “just in a few lines” fromAlbert’s result, as has been said in [Zel73]. However, by looking more closely intothe matter it turns out that “theorem 13” is based on “theorem 23” whose proof usesthe same arguments as Brauer–Noether (as we have said above already). Conversely,“theorem 13” is immediate if one uses the chain of arguments given by Brauer andNoether (p. 14). In other words: both methods, that of Brauer–Noether and that ofAlbert, are essentially the same, the differences concerning non-essential details only.

It is not an unusual mathematical story: A major result is lurking behind thescenes, ready to be proved, and more than one mathematician succeeds. So thishappened here too.

1.8.4 The priority question

Let us return to November 9, 1931, the date when Hasse found the last steps in theproof of the Main Theorem. We have seen in Section 1.3 that Hasse immediatelyinformed Noether and Brauer about it. But he also informed Albert. Hasse’s letter toAlbert was dated November 11. This letter crossed paths withAlbert’s of November 6.Albert received it on November 26, and his reply is dated November 27. Therehe congratulated Hasse to the “remarkable theorem you have proved”. But in themeantime, he added, he had already obtained results which also could be used toprove the Local-Global Principle; they are contained in a paper (which Albert had notmentioned to Hasse before) in the Bulletin of the AMS [Alb31a]. Albert pointed outthat it had been submitted on September 9 and the issue of the Bulletin had alreadybeen delivered in October. It contained (among other results) the “theorem 13” whichwe have just discussed. Albert writes:

As my theorems have already been printed I believe that I may perhapsdeserve some priority of your proof. I may say, however, that the re-markable part of your proof for me is the obtaining of the cyclic field. Iof course knew your theorem 3.13.

It is not clear what theorem Albert refers to when he cites “3.13”. Neither the Brauer–Hasse–Noether paper nor Hasse’s American paper contains a theorem with this num-ber. From an earlier part of Albert’s letter it seems probable that “theorem 3.13” maystand for the Norm Theorem (see Section 1.4.1). But in Hasse’s American paper thishas the number (3.11); thus it may have been just a misprint on the side of Albert.

When Albert speaks of “obtaining of the cyclic field” to be the “remarkable part”of Hasse’s proof then he refers to the Existence Theorem which we have discussed inSection 1.5.2. This shows that Albert immediately saw the weak point of the Brauer–Hasse–Noether paper. For, the existence of the required cyclic field was not proved


in the Brauer–Hasse–Noether paper. Albert expressed his hope that the existence ofthe cyclic field

… will be made clear when you publish your proof. In all my workon division algebras the principal difficulty has been to somehow find acyclic splitting field. This your p-adic method accomplishes.

Probably Albert meant not just some cyclic splitting field but a splitting field of degreeequal to the index of the division algebra. As we have seen in Section 1.5.2 the storyof this Existence Theorem is not quite straightforward. But if he really wanted only tofind a cyclic splitting field of unspecified degree, then the proof of the relevant weakexistence theorem was contained in Hasse’s paper [Has33a], as we had mentionedalready in Section 1.5.4.

Hasse responded to Albert’s wish for “some priority” by extending the Albert-footnote a second time, adding a third part where he stated what Albert had writtento him in his letter of November 26. But when Hasse wrote in the footnote thatAlbert’s paper is “currently printed” (“im Druck befindlich”), then this may havebeen a misinterpretation of Albert’s text in the letter, which reads as follows:

The part of the proof which you attribute to Brauer and Noether is alreadyin print.

Obviously, Albert meant that the paper has already appeared (namely in October) andthus is available in printed form. The German translation of “in print” would be “imDruck”. But in German, if it is said that some paper is “im Druck” then the meaningis that the paper is “in press”, i.e., in the process of being printed. This may have ledHasse to the wrong translation of Albert’s text. In any case, neither Hasse nor Noethernor Brauer had yet seen Albert’s Bulletin paper. If indeed it had appeared in Octoberin the U.S.A. then it was not yet available in German libraries in the beginning ofNovember.

We conclude that Albert’s Bulletin paper and the Brauer–Hasse–Noether paperhad been written independently of each other. On the other hand, as Albert pointsout correctly, his results in his Bulletin paper can be used to prove the Local-GlobalPrinciple for algebras and hence indeed constitute an “independent share” in the proofof the Main Theorem. We have discussed this in the foregoing section.

Responding to Albert’s wish for priority, Hasse did two more things besides ex-tending the Albert-footnote a second time. First, he sent Albert a copy of the proofsheets of the Brauer–Hasse–Noether paper, so he could check in particular the actualtext of the Albert-footnote. Secondly, Hasse suggested that they write a joint paper,to be published in the Transactions, documenting the sequence of events which led tothe proof of the Main Theorem, on the Albert side as well as on the side of Brauer–Hasse–Noether. Albert should write up the article. In his letter of January 25, 1932Albert reported:


I have finally found time to write up the article by both of us A determi-nation of all normal division algebras over an algebraic number field forthe Transactions. I gave a historical sketch of the proof, my short proofand a slight revision (to make it more suitable for American readers) ofyour proof.

This 5-page paper [AH32] appeared in the same volume as Hasse’s American paper[Has32c]. It documents for the mathematical community the chain of events whichwe have extracted here from the Albert–Hasse correspondence.

Irving Kaplansky in his memoir on Albert [Kap80] writes:

In the hunt for rational division algebras, Albert had stiff competi-tion. Three top German algebraists (Richard Brauer, Helmut Hasse,and Emmy Noether) were after the same big game … It was an unequalbattle, and Albert was nosed out in a photo finish. In a joint paper withHasse published in 1932 the full history of the matter was set out, andone can see how close Albert came to winning.

This comparison with a competitive sports event reads nicely but after studying theAlbert–Hasse correspondence I have the impression that it does not quite reflect thesituation. In my opinion, it was not like a competitive game between Albert on theone hand and the trio Brauer–Hasse–Noether on the other. Instead it was teamwork,first among Brauer, Hasse, Noether and then, starting March 1931, Albert joined theteam as the fourth member. Within the team, information of any result, whether smallor important, was readily exchanged with the aim to reach the envisaged commongoal. If a comparison with a sports event is to be given, then perhaps we can look atit as a team of mountaineers who joined to reach the top. The tragedy was that oneof the team members (Albert) in the last minute lost contact with the others (becausecommunication was not fast enough) and so they reached the summit on differenttrails in divided forces, 3 to 1.

Nevertheless, Albert in this situation was upset, which of course is quite under-standable. Zelinsky in [Zel73] writes that “Albert was hurt and disappointed by thisincident.” I may be allowed to cite Professor Zelinsky’s answer when I asked himabout his memories of Albert.

Perhaps my use of the word “hurt” was injudicious, since besides themental pain that Albert must have felt, the word could connote feelingsthat he had been taken advantage of, that his correspondence with Hassewas used without due consideration. I have no evidence that he feltthat way. You are correct, he was content at last with the resolution ofthe priority questions. And by the time I knew him, he had becomeestablished as a leading mathematician in his own right, which surelyaffected his attitude toward events of the previous decade.


We believe that, indeed, one can infer from the letters of Albert that he was contentwith the solution of the priority question as offered by Hasse. The exchange of letterswith mathematical results continued in a friendly tone. We have the impression thatfrom now on the tone of the letters was even more open and free. At one time Albertcriticised an error which Hasse had made in a previous letter, and at another occasionHasse pointed out an error of Albert’s – in an open and friendly way. At the end of hisletter of January 25, 1932 Albert added a somewhat remarkable postscript as follows:

Permit me to say I did not believe it possible for mere correspondence toarouse such deep feelings of friendship and comradeship as I now feelfor you. I hope you reciprocate.

We do not know Hasse’s answer. In Albert’s next letter (April 1, 1932) he thanksHasse for sending “photographs and books”. Moreover we read in this letter:

I am very pleased to have been asked to write a report on linear algebrasfor the Jahresbericht. I shall certainly accept this kind of proposition … Ishall study your report and try to understand better precisely what typeof report you wish me to write.

Here, “your report” means Hasse’s class field report [Has30a].It seems that Hasse wished to revive the long tradition of the Jahresbericht of

the DMV, to publish reports on recent developments in fields of current interest. 81

However, just at the same time there arose a stiff competition, in the concept ofSpringer-Verlag to initiate a series of books called Ergebnisse der Mathematik. Wedo not know whether it was this competition or there were other reasons that theJahresbericht report series was not continued by the DMV. In fact, Hasse’s report[Has30a] turned out to be the last one in this series and the plan to publish Albert’sreport in this series was not realized.82

In some of the next letters which followed, Albert still tried to convince Hasse thatthe arguments in the Brauer–Hasse–Noether paper included unnecessary complica-tions whereas his chain of arguments was, in his opinion, shorter and more lucid. Thisis understandable because as the author he was more familiar with his own version.But we have the impression that Hasse was not convinced, although he indeed hadhigh respect for Albert’s achievements which had been obtained in a relatively short

81One of the best known such reports is Hilbert’s Zahlbericht 1897.82As a side remark we may mention that already in 1931 Albert had been asked to write a survey on the theory

of algebras, just by the competitor of the Jahresbericht series, the newly inaugurated Springer series Ergebnisseder Mathematik. But after a while this proposal was withdrawn by Otto Neugebauer, the editor of the series.According to Albert (letter of December 9, 1931) Neugebauer wrote that it had been arranged with Deuringto write a survey on “Hypercomplex Systems”, and that he (Neugebauer) had discovered just now only thatthis is the same subject as “Algebras”. Deuring’s book appeared 1934 with the title “Algebren”. Actually, weknow that Neugebauer had first approached Emmy Noether to write such a survey but she declined and, instead,recommended her “best student” Deuring for this task. As to Deuring’s book, see also [Roq89]. – It seems thatAlbert after these experiences decided to write his book on the Structure of Algebras [Alb39] independently andpublish it elsewhere.


time after their correspondence started. Sometimes Hasse scribbled some marginalnotes on Albert’s letters as reminders for his reply, and from this we can extract atleast some marginal information about Hasse’s letters to Albert. On April 1, 1932Albert wrote that he believed

… your whole Transactions paper could be simplified considerably ifthis reduction had been made to begin with. Of course it is a matter ofpersonal taste and you may even yet not agree.

Here, Albert has in mind the reduction from arbitrary algebras to those of primepower index by means of Brauer’s product theorem which he, Albert, had discoveredindependently.

To the second sentence we find the note “yes!” on the margin of the letter, writtenby Hasse’s hand. This seems to imply that Hasse’s personal taste was somewhatdifferent. After all, as we have seen in Section 1.4.2, Hasse’s first draft containedthis reduction to prime power index, and it was Emmy Noether who threw this awaybecause it was unnecessary. – The first sentence is commented by “no! (class fieldtheory!)”. Here Hasse refers to the close connection of the theory of algebras toclass field theory – something which was outside of the realm of interest of Albert,except that he admitted class field theory as a means to prove theorems on algebras,if necessary. See also Section 1.6.3.

1.8.5 Remarks

It seems that Hasse had invited Albert to visit him in Germany. In his letter of June 30,1931 Albert wrote:

I hope that in perhaps two years I may visit Germany and there see youand discuss our beautiful subject, linear algebras.

On November 27, 1931 the plans had become more specific:

I am very glad that you are interested in the possibility of my visitingyou. I hope that I will be able to leave Chicago on Sept 1, 1933 to returnhere not later than Dec 31, 1933. I do not believe I can make the tripbefore that time.

Due to the disastrous political events which took place in Germany in the year1933 these plans could not be realized. Instead, Albert applied for and received anappointment at the Institute for Advanced Study in Princeton for the academic year1933/34. There are two letters from Albert in Princeton to Hasse which are preserved.They do contain interesting material but this is not immediately connected with theMain Theorem, hence we will not discuss it here.

While Albert was in Princeton in 1933/34 he met two-thirds of the German team,Richard Brauer and Emmy Noether, who had been forced to leave their country.


On the relationship between Albert and Brauer, Mrs. Nancy Albert, the daughter ofA.A.Albert, reports:83

I have news about Brauer from letters written to my parents. WhenBrauer arrived in America from Germany, he spoke little English, andwas rather traumatized from all that he had been through. He stoppedat Princeton in 1933. My father took Brauer under his wing, made himfeel welcome, and took him to meet Wedderburn. Later, my father puttogether a large mathematical conference in Chicago, where the Albertshosted a large dinner party, and the Brauers became good friends of myparents. Their relationship continued the remainder of my father’s life.

Mathematically, however, Albert’s and Brauer’s work went in somewhat differentdirections. Albert continued to work on algebras, including more and more non-associative structures. Brauer concentrated on group theory and representations.Most of the work on finite simple groups and their classification can be traced tohis pioneering achievements, and he advanced to “one of the leading figures on theinternational mathematical scene” (J.A. Green).

About the Princeton contact of Albert with Noether we have the following infor-mation. In an undated letter to Hasse from Princeton, probably written in January1934, Albert writes:

I have seen R. Brauer and E. Noether. They passed through here andstayed a short while.

And on February 6, 1934:

E. N. speaks here tomorrow on Hypercomplex numbers and Numbertheory.

Emmy Noether herself, in her letters to Hasse, is somewhat more detailed about themathematical life in Princeton. In a letter of March 6, 1934 she writes the followingreport, and we note that Albert is mentioned:

… habe ich seit Februar einmal wöchentlich eine Vorlesung in Prince-ton angefangen – am Institut und nicht an der “Männer”-Universität, dienichts Weibliches zuläßt … Ich habe mit Darstellungsmoduln, Gruppenmit Operatoren angefangen; Princeton wird in diesem Winter zum er-stenmal, aber gleich gründlich, algebraisch behandelt. Weyl liest auchDarstellungstheorie, will allerdings zu kontinuierlichen Gruppen überge-hen. Albert, in einem “Leave of absence” dort, hat vor Weihnachtenetwas hyperkomplex nach Dickson vorgetragen, zusammen mit seinen“Riemann matrices”. Vandiver, auch “Leave of absence”, liest Zahlen-theorie, zum ersten Mal seit Menschengedenken in Princeton. Und von

83Personal communication.


Neumann hat – nach einem Überblick von mir über Klassenkörperthe-orie im mathematischen Klub – gleich zwölf Exemplare Chevalley alsLehrbuch beordert (Bryn Mawr soll auch etwas davon bekommen!).Dadurch erfuhr ich auch, daß Ihre Ausarbeitung ins Englische über-setzt wird, jetzt hoffentlich in genügend vielen Exemplaren – daraufhatte ich die Leute schon gleich im Herbst gehetzt. Ich habe wesentlichResearch-fellows als Zuhörer, neben Albert und Vandiver, merke aberdaß ich vorsichtig sein muß; sie sind doch wesentlich an explizites Rech-nen gewöhnt, und einige habe ich schon vertrieben!

… I have, since February, started a lecture in Princeton once a week –at the Institute and not at the “men’s”-university which does not admitanything female …At the beginning I have started with representationmodules, groups with operators. This winter Princeton is treated alge-braically, for the first time but quite thoroughly. Weyl also lectures aboutrepresentation theory but will soon switch to continuous groups. Albert,in a “leave of absence” there, has last year lectured on something hyper-complex in the style of Dickson, together with his “Riemann matrices”.Vandiver, also “leave of absence”, lectures on number theory, the firsttime in Princeton since time immemorial. And after I had given a sur-vey on class field theory in the Mathematics Club, von Neumann hasordered twelve copies of Chevalley as a textbook (Bryn Mawr also shallget a copy). On this occasion I was told that your Lecture Notes willbe translated into English, now hopefully in sufficiently many copies –I had recommended this already in the fall. My audience consists es-sentially of research fellows, besides Albert and Vandiver, but I noticedthat I have to be careful; these people are used to explicit computations,and some of them I have already driven away!

We can safely assume that Albert was not one of the dropouts from Noether’s course.He knew about the importance of Emmy Noether’s viewpoint on algebra and onthe whole of mathematics. Noether’s ideas have often been described and so wewill not repeat this here.84 But at the time we are considering, Noether’s ideashad not yet penetrated mathematics everywhere. Albert himself had his training withDickson, and his papers in those first years of his mathematical activity were definitely“Dickson style”. It was only gradually that Albert started to use in his papers the“Modern Algebra” concepts in the sense of Emmy Noether and van der Waerden. In1937 Albert published the book Modern higher algebra [Alb37] which was a studenttextbook in the “modern” (at that time) way of mathematical thinking.85

Albert explicitly stated that his textbook was meant as an introduction to themethods which will be used in his forthcoming book on algebras. That second book

84See, e.g., Hermann Weyl’s obituary address in Bryn Mawr 1935 [Wey35].85The book was refereed in the Zentralblatt by Helmut Hasse.


appeared in 1939 [Alb39] with the title Structure of Algebras, and it was also writtenwith the viewpoint of “Modern Algebra”.86 It seems to us that to a large degree thiswas a direct consequence of Albert being exposed at Princeton to Emmy Noether’sinfluence.

In the preface to his book Albert says:

The theory of linear associative algebras probably reached its zenithwhen the solution was found for the problem of determining all rationaldivision algebras. Since that time it has been my hope that I mightdevelop a reasonably self-contained exposition of that solution as wellas of the theory of algebras upon which it depends and which containsthe major portion of my own discoveries.

We do not intend here to give a review of Albert’s book which, after all, is wellknown and has become a classic. It is our aim here to point out that to a large extentthe book is the outcome of his participation in the team together with Hasse, Brauerand Noether – notwithstanding the fact that the book contains also other aspects ofthe theory of algebras, e.g., Riemann matrices and p-algebras.

But when Albert said that “the theory of algebras had reached its zenith” withthe Main Theorem then we cannot agree. Since then a number of highly importantresults have been established, and the theory is still flourishing.

1.9 Epilogue: Käte Hey

In the history of mathematics we can observe not infrequently that after an importantresult has been found, it is discovered that the very same result, in more or less explicitform, had been discovered earlier already. This happened also to the Local-GlobalPrinciple for algebras which is the basis for Hasse’s proof of the Main Theorem.

On January 26, 1933, one year after the appearance of the Brauer–Hasse–Noetherpaper, the editor of the Hamburger Abhandlungen received a manuscript of a paper[Zor33] which begins as follows:

Die Theorie der �-Funktion eines Schiefkörpers ist von Fräulein K. Heyin ihrer Dissertation (Hamburg 1929 ) eingehend entwickelt worden: ichmöchte in dieser Note auf die arithmetischen Konsequenzen, die dort ausder analytischen Theorie gezogen werden, aufmerksam machen und sieauf Grund einiger Korrekturen und Abrundungen als

neuen Beweis eines Hauptsatzes für Algebren sowiedes allgemeinen quadratischen Reziprozitätsgesetzes

86This book was refereed in the Zentralblatt by W. Franz, a former Ph. D. student of Hasse.


erkennen lassen. Der in Frage stehende Hauptsatz ist die Grundlage fürdie Herleitung des Reziprozitätsgesetzes mit nichtkommutativen Mit-teln; seine independente Begründung ist also methodisch wichtig.

The theory of the �-function of a skew field has been developed in detailby Miss K. Hey in her dissertation (Hamburg 1929). In the present noteI would like to draw attention to the arithmetic consequences which arederived there, so that after some correction and streamlining they arerecognized as a

new proof of a main theorem on algebrasand of the general quadratic reciprocity law.

The said main theorem on algebras is the basis for deriving the reci-procity law with non-commutative methods; therefore its independentfoundation is important for methodical reasons.

The “main theorem” which is meant here is not quite the Main Theorem of Brauer–Hasse–Noether but the Local-Global Principle as formulated in Section 1.4. The“general quadratic reciprocity law” is extra mentioned by the author because it followsdirectly from Käte Hey’s treatment in the case of quaternion algebras. In the nextsentence however, “reciprocity law” meansArtin’s reciprocity law; to derive this fromthe Main Theorem one had to follow Hasse’s method as explained in Section 1.6.

The author of this article was Max Zorn, a former Ph. D. student of Artin inHamburg.87 He had been the second Ph. D. student of Artin, the first one had beenKäte Hey whose thesis he is referring to in his note. She received her degree in 1927.88

Her thesis [Hey29] had never been published in a mathematical journal but it wasprinted, and was distributed among interested mathematicians. We know that Hasseand Emmy Noether each owned a copy, perhaps Richard Brauer too. The thesis wasrefereed in the Jahrbuch für die Fortschritte der Mathematik, vol. 56.

The aim of Hey’s thesis was to extend the known methods of analytic numbertheory to division algebras instead of number fields – in particular the methods ofHecke which lead to the functional equation of zeta functions. She defined the zetafunction �D.s/ of a division algebra D whose center K is an algebraic number field.But she considered only the finite primes p of K. If that function is supplementedby factors corresponding to the infinite primes of K (which today is the standardprocedure) then the analytic treatment of that extended function, including its func-tional equation, shows that, if compared with the zeta function �K.s/ of the center, itadmits two poles (if D ¤ K), which in some way correspond to primes p which are

87Zorn (1906–1993) received his Ph. D. 1930 with a paper on alternative algebras. In 1933 he was forcedby the Nazi regime to leave Germany. His name is known to the mathematical community through his “Zorn’sLemma”.

88Käte Hey (1904–1990) left the university some time after she had obtained her degree, then she became ateacher at a gymnasium. More biographical details can be found in [Lor05] and [Tob97].


ramified in D, i.e., at which Dp does not split. Indeed, the existence of such primesis the content of the Local-Global Principle. Hey used Artin’s paper [Art28b] on thearithmetic of algebras, and also the full arsenal of analytic number theory known atthat time which centered around Hecke’s work. Deuring says in [Deu35]:

Dieser Beweis des Hauptsatzes über Algebren ist gleichsam die stärk-ste Zusammenfassung der analytischen Hilfsmittel zur Erreichung desZieles.

In a way, Hey’s proof of the Main Theorem represents the strongestconcentration of analytic tools to reach the aim.

For a discussion of Hey’s thesis and Zorn’s note we refer to the recent essay[Lor05] by Falko Lorenz.

Hey’s thesis is considered to be difficult to read. It seemed to be generally knownat the time, at least among the specialists, that Hey’s thesis contained errors. But Zornpoints out how those errors could be corrected in a quite natural and straightforwardmanner.

Thus if Hasse (or Noether, or Brauer, or Albert) had known this earlier, then theproof of the Main Theorem could have been completed earlier. It is curious that Hey’sthesis had not been mentioned in the correspondence of Hasse, not with Artin, notwith Noether, Brauer or Albert. At least not before Zorn’s note became known.89

Later, when Hasse and Noether discussed how much analysis should and couldbe used in class field theory, Hasse wrote to her (letter of November 19, 1934):

… wenn man schon einmal Analysis zur Begründung der Klassenkör-pertheorie braucht, … man dann die Kanone auch gleich auf …den Nor-mensatz, das Summen-Theorem für die Invarianten und den Satz vonden überall zerfallenden Algebren richten soll. Konsequenz: Man ver-wende die Heysche Kanone …und dann Rückschuss auf die klassischeKlassenkörpertheorie wie zu Ihrem 50. Geburtstag.

… if analysis is to be used in the foundation of class field theory … thenone should aim with Hey’s cannon at the norm theorem, the sum formulafor the invariants of algebras, and the theorem on algebras splitting ev-erywhere. Consequently, one should use Hey’s cannon … and then aimbackwards to class field theory in the classical sense, like on your 50thbirthday.

89Emmy Noether got to know Zorn’s note some time in winter 1932/33. She was so impressed that shesuggested to two of her Ph.D. students to continue work in that direction. One of those students was Ernst Wittwho in his thesis transferred Hey’s results to the function field case. The other student was Wolfgang Wichmannwho presented a much simplified proof of Hey’s functional equation of the zeta function of a division algebra,however up to a ˙ sign only.


Here, the reference to Noether’s 50th birthday is to be read as the reference to Hasse’spaper [Has33a] which he had dedicated to her on the occasion of that birthday; wehave mentioned it several times in this paper.

In his letter Hasse pointed out that no one at that time had been able to develop classfield theory without using methods from analytic number theory. His own approachin the “birthday paper” [Has33a] is based on the Hilbert–Furtwängler Norm Theoremwhich in turn was proved using analytic methods of zeta functions of number fields.But soon after, Chevalley [Che35] succeeded to give a foundation of class field theoryfree from analysis; see also [Che40].

Chapter 2

The remarkable career of Otto Grün

From FLT to finite groups. The remarkable career of Otto Grün.

Jahresbericht der Deutschen Mathematiker Vereinigung 107 (2005), 117–154

(Section 2.9 has been added.)

2.1 Introduction 772.2 The first letters: FLT (1932) 782.3 From FLT to finite groups (1933) 852.4 The two classic theorems of Grün 902.5 Grün meets Hasse (1935) 982.6 The Burnside problem (1939) 1042.7 Later years (after 1945) 1092.8 Epilogue 1132.9 Addendum 115

2.1 Introduction

Students who start to learn the theory of finite groups will soon be confronted withthe theorems of Grün, at least with Grün’s “first” and “second” theorem, and withits generalizations.. These theorems found their way into group theory textbooksimmediately after their publication in the mid 1930s, with the comment that they areof fundamental importance in connection with the classical Sylow theorems. Butlittle if anything is known about the mathematician whose name is connected withthose theorems.

Recently, scanning through the legacy of Helmut Hasse which is kept at theUniversity Library in Göttingen, I found 50 letters which were exchanged betweenHasse and Grün, from 1932 to 1972. Hasse is known to have had an extendedcorrespondence, freely exchanging mathematical information with his colleagues.Thus at first sight, I was not really surprised to find the name of Otto Grün amongHasse’s many correspondence partners.

78 2 The remarkable career of Otto Grün

But while reading these letters there unfolded to me much more than just mathe-matical information, namely the remarkable and fascinating story of a mathematician,quite rare in our time, who was completely self-educated, without having attendeduniversity, and nevertheless succeeded, starting at age 44, to give important contri-butions to mathematics, in particular to group theory.

I am writing this article in order to share this discovery with other interestedmathematicians. But I would like to make it clear that this is not meant to be a completebiography of Otto Grün. This article comprises mainly what we can conclude fromthe correspondence files of Hasse and some secondary sources, with emphasis on thegenesis of Grün’s main theorems. Perhaps a more detailed search of other sourcescould bring to light more facets of Grün’s personality and work.

Acknowledgement. I had sent a former version of this article to a number of peoplewho (like myself) had met Grün and still remember him. I would like to thank allcolleagues for their interest and for their various comments on the work and thepersonality of Grün. In particular I would like to thank B. Huppert and W. Gaschützfor their help concerning the group theoretic part of Grün’s work. It seems to me thata more detailed survey of Grün’s role in the development of group theory would beinteresting and worthwhile. Last but not least I would like to thank the referee forseveral well founded comments.

2.2 The first letters: FLT (1932)

2.2.1 Grün and Hasse in 1932

Little is known about the early life of Grün. In his vita which he wrote in 1955 weread:

Ich bin am 26. Juni 1888 zu Berlin geboren, besuchte das Friedrich-Werdersche Gymnasium zu Berlin, das ich 1908 mit dem Reifezeugnisverließ. Zunächst widmete ich mich dem Bankfach, nahm am erstenWeltkriege teil, ohne Schäden davonzutragen, und war weiterhin kauf-männisch tätig.

I was born on June 26, 1888 in Berlin. I attended the Friedrichs-WerderGymnasium in Berlin until 1908. Then I worked in the banking business,participated in the first world war without being injured, and afterwardscontinued to work in a commercial job.

It is not known what kind of job this had been.1 Grün continued:

1But see our Addendum at the end of this article.


Da ich stets lebhaftes Interesse für mathematische Fragen hatte, beschäf-tigte ich mich nebenbei wissenschaftlich und kam auf diese Art zu einemBriefwechsel mit dem berühmten Algebraiker Helmut Hasse …

All the time I had strong interest in mathematical problems, and in myspare time I occupied myself with scientific problems.2 In the courseof this activity there started an exchange of letters with the famous al-gebraist Helmut Hasse …3

The first letter of Otto Grün to Hasse is dated March 29, 1932, from Berlin. At thattime Grün was (almost) 44 years old.

Otto Grün in 1957

Helmut Hasse, 10 years younger than Grün, at that time was professor of Mathe-matics at the University of Marburg (since 1930) as the successor of Kurt Hensel. Theyears in the late twenties and early thirties are to be regarded as the most fruitful pe-riod in Hasse’s mathematical life. Hasse had completed the last part of his class fieldtheorys report [Has30a], he had proved (with Richard Brauer and Emmy Noether) theLocal-Global Principle for simple algebras [BHN32], he had determined the structureof cyclic algebras over a number field [Has32c], he had discovered local class field

2Here and in the following we use our own free translation of German text into English.3 I have found the vita which starts with the cited sentences, in the archives of the University of Würzburg

where Grün had a teaching assignment (“Lehrauftrag”) during the years from 1954 to 1963 (see Section 2.7.3below). It is datedAugust 2, 1955. I do not know the occasion for which Grün had presented this to the university.Probably it was connected with his teaching assignment. – I am indebted in particular to Hans-Joachim Vollrathfor his help to obtain access to the Würzburg archives.


theory [Has30c], given a new foundation of the theory of complex multiplication[Has27b], [Has31b], and more. In Hasse’s bibliography we have counted more thanfifty papers in the period from 1926 to 1934. In March 1932, when he received Grün’sfirst letter, he had just completed his seminal paper dedicated to Emmy Noether onher 50th birthday [Has33a], where he presented a proof of Artin’s Reciprocity Lawin the framework of simple algebras and at the same time determined the structureof the Brauer group over a number field. Now he was preparing his lecture courseon class field theory which he was to deliver in the summer semester of 1932, thenotes of which [Has33b] would be distributed widely and would influence the furtherdevelopment of class field theory. One year later, in March 1933, Hasse would provethe Riemann hypothesis for elliptic function fields.

It seems remarkable that in the midst of all this activity, Hasse found the time todeal carefully with the letters of Otto Grün, whom he had never heard of before. Hassehad the strong viewpoint that every letter from an amateur mathematician representsan unusual interest in mathematics by the sender and, hence, has to be taken seriously.And so he did with Grün’s letter, thereby discovering that the sender was not one ofthe usual Fermatists but, despite his lack of formal mathematical education, wasunusually gifted and had a solid mathematical background.

2.2.2 Vandiver’s conjecture and more

Grün’s first letter begins as follows:

Sehr geehrter Herr Professor! Ich habe aus Ihren Arbeiten die TakagischeKlassenkörpertheorie kennengelernt. Ich glaube nun, auf dieser Grund-lage zeigen zu können, daß auch im irregulären Körper k.�/ der `-tenEinheitswurzeln der 2-te Faktor der Klassenzahl nie durch ` teilbar seinkann. Darf ich Ihnen vielleicht hier ganz kurz den Beweis skizzieren, zu-mal da ich als reiner Amateurmathematiker denselben doch nicht veröf-fentlichen würde.

Dear Herr Professor! From your papers I have learned Takagi’s classfield theory. I believe that on this basis I can show that also in theirregular field Q.�/ 4 of the `-th roots of unity the 2nd factor of the classnumber can never be divisible by `. May I sketch briefly the proof sinceanyhow, as an amateur mathematician, I am not prepared to publish it.

By “class number” Grün means the number of ideal classes of the `-th cyclotomicfield Q.�/ where � denotes a primitive `-th root of unity, ` being an odd prime. It iswell known since Kummer [Kum50] that the class number h of Q.�/ has a product

4Grün writes k.�/ (in conformity with the older notation) where we have written Q.�/ (which is today’snotation). In the interest of the reader we shall freely change notations from the original, whenever it seemsappropriate for better understanding,


decompositionh D h1h2

where the second factor h2 equals the class number of the maximal real subfieldQ.�C��1/. The first factorh1 is a positive integer, called the “relative class number”.5

The prime `, or the field Q.�/, is called “regular” if the class number h is not divisibleby `. One of the monumental achievements of Kummer [Kum50] was the discoverythat FLT holds for a regular prime number `, which is to say that the diophantineequation

x` C y` C z` D 0

is not solvable in integers x; y; z ¤ 0.If ` is regular then, of course, both factors h1 and h2 are not divisible by `. If ` is

irregular then it was known to Kummer already that h1 is divisible by `, but nothingmuch was known about h2. Now Grün claimed that h2, even in the irregular case, isnot divisible by `. This statement is today known as “Vandiver’s conjecture”, and itis considered quite important with respect to the structure of cyclotomic fields.6

Thus in effect Grün claimed to have proved Vandiver’s conjecture, although hedid not mention Vandiver in his letter. Most probably he was not aware at that timeof Vandiver’s work.

Hasse replied on April 1, 1932 already, three days after Grün had dispatched hisletter. We do not know the text of Hasse’s letter7 but from Grün’s answer we candeduce that Hasse had pointed out the proof to be erroneous. Grün wrote on June 27,1932:

Gegen Ihre Bedenken kann ich natürlich nichts einwenden; der Beweisist eben in der vorliegenden Form mißglückt.

Of course there cannot be any objection against your doubts. Thus myproof has not been successful in this form.

In fact, Vandiver’s conjecture has not been proved or disproved until today, despitestrong efforts by many mathematicians.8

5The terminology “first” and “second” factor of the class number is generally used in the literature. But Hassein his book [Has52] says that the inverse enumeration would be more natural: h2 should be called the “first” andh1 the “second” factor. In later letters (1957/58) Grün uses therefore this inverse terminology. Hasse himself in[Has52] writes h� for h1 and h0 for h2. Ribenboim [Rib79] writes hC for h2.

6I am indebted to Franz Lemmermeyer for pointing out to me the paper [Van41] in which Vandiver expresseshis “hope” that h2 is always prime to `. Ribenboim [Rib79] remarks that Vandiver’s conjecture is already statedin a letter of Kummer to Kronecker, dated December 28, 1849.

7Quite generally, the letters from Grün to Hasse are preserved in the Hasse legacy, whereas many of theletters from Hasse to Grün have to be considered as lost. Only in later years, in case the letters were writtenby typewriter, Hasse used to make a carbon copy for himself and so his letters are preserved too. But this wasnot the case for his early letters to Grün which were handwritten. In most cases however, by interpolating fromGrün’s replies we can deduce approximately what Hasse had written.

8By now the conjecture has been verified for all odd primes ` < 12 � 106 (communication by FranzLemmermeyer).


But Hasse had not been content just to point out the error in Grün’s proof. Hehad added some comments for further work. Maybe he also recommended to Grünsome of the relevant literature. For, Grün wrote in his second letter to Hasse (June27, 1932) the following:

… glaube ich aus Ihrem Hinweis eine Folgerung für die FermatscheBehauptung ziehen zu können, die ich Ihnen gern mitteilen möchte …Wenn x` C y` C z` D 0 in rationalen Zahlen x, y, z und etwa x durch` teilbar, yz prim zu ` ist, so muß der zweite Faktor der Klassenzahldurch ` teilbar sein.

… in view of your comments I believe that I can derive the following re-sult towards Fermat’s Last Theorem which I would like to communicateto you … If x` C y` C z` D 0 is solvable in nonzero rational integersx, y, z and x is divisible by ` while yz is not divisible by ` then thesecond factor of the class number is divisible by `.

In dealing with the Fermat equation one usually distinguishes two cases: In the “firstcase” one assumes that none of x, y, z is divisible by `. In the “second case” one ofthem, say x, is divisible by ` whereas y, z are not. Thus Grün’s claim says in effect:

If the second class number factor h2 is not divisible by ` then the Fermatequation is not solvable in the so-called second case.

And he sketched a proof of this result. But again, there was an error which Hassepointed out to him in a letter two days later.

We should keep in mind that Grün had not received any formal mathematicaleducation; mathematically he was completely self educated and this was the firstoccasion where he could discuss his ideas with a competent mathematician. Thesubject required a high level of sophistication, and after all he had no training inpresenting mathematical ideas. Thus the failure of his first attempts to producea consistent proof is understandable. He was lucky to have found Hasse as hiscorrespondence partner who, it seems, had recognized the mathematical capacity ofthe author of those letters.

After some more months, on September 28, 1932 Grün wrote again. He saidthat he had indeed observed the difficulty which Hasse had pointed out to him buthad erroneously assumed that this could be handled by the methods of Kummer.Nevertheless, he now presented a correction of his result, namely with an additionalhypothesis concerning certain divisibility properties of Bernoulli numbers.

The sequence of Bernoulli numbersBn can be defined as the coefficients appearingin the power series expansion

x

ex � 1 DXn�0

Bn

xn

nŠ:


These Bn are certain rational numbers which are well known to be connected withthe structure of the cyclotomic field Q.�/. Kummer had used the Bn to formu-late a necessary and sufficient criterion for ` to be regular. Namely, the numbersB2; B4; : : : ; B`�3 should not be divisible by `. 9 Kummer had also discussed theirregular case to some extent, and there too he had given sufficient criteria for thevalidity of FLT.

Now Grün’s additional hypothesis reads as follows:

Es mögen zwar beliebig viele Bernoullische Zahlen Bi mit i < ` � 1

durch ` in erster Potenz teilbar sein, jedoch gelte für keine von ihnenB`i 0 mod `3 bei geradem i .

Arbitrary many Bernoulli numbers Bi with i < ` � 1 may be divisibleby `, but for none of them we have B`�i 0 mod `3 with i even.

It was well known, already to Kummer, that this hypothesis implies certain structuralproperties of the group of units of Q.�/. Grün showed that it is sufficient (in additionto the hypothesis that the second class number factor h2 is not divisible by `) todeduce that the Fermat equation for exponent ` has no solution in the second case.10

This time Hasse did not find an error in Grün’s proof. But he wanted to be surethat Grün’s result was new. Perhaps Hasse remembered a paper by Vandiver [Van29]which in fact contained Grün’s above cited result. But Grün’s computations yieldedat the same time a somewhat more general result than we have cited above, showingthe impossibility not only for the Fermat equation in the second case, but also forcertain other diophantine equations within the cyclotomic field Q.�/, going beyondVandiver’s results. Hence, even if Grün’s result for the Fermat equation was known,perhaps his more general result was new?

Thus Hasse proposed that Grün should write to Vandiver at the University ofTexas who was considered to be a specialist on those problems. Grün replied that heis afraid not to know the proper mathematical terminology in English language, andanyhow he does not know the address of Vandiver. Upon this Hasse himself wrote toVandiver on behalf of Grün. Since several years Hasse had exchanged reprints withVandiver and, as can be seen from the correspondence between the two, the latter hadvisited Hasse at least twice, once in Halle and another time in Marburg.11

Vandiver replied in a letter of November 14, 1932:

9If the index n > 1 is odd then Bn D 0. Because of this, the enumeration of the Bernoulli numbers issometimes changed, i.e., writing Bn instead of B2n for n � 1. But we will keep the notation as given by thedefinition above.

10Ribenboim [Rib79] (p. 188) says erroneously that Grün’s result refers to the first case.11Vandiver too, like Grün, did not have a formal mathematics education. In the biography of Vandiver

(1882–1973) in [Leh74] we read: “This remarkable man … was self-taught in his youth and must have had littlepatience with secondary education since he never graduated from high school.” However, already with 22 yearsVandiver wrote his first mathematical paper whereas Grün was 47 when his first paper appeared.


The two theorems you mention appear to be quite new. The first oneseems to be a modification and extension of the Theorem I of my paperof the year 1929 in the Transactions A. M. S.

Here Vandiver cites his paper [Van29].This must have been sufficient for Hasse. Perhaps he was aware of the fact that one

year earlier, in 1931, Vandiver had been awarded from the American MathematicalSociety the highly prestigious Cole Prize in number theory for his work on FLT, inparticular for his paper in the Transactions which Vandiver was citing in his letter. IfGrün’s result was an extension of Vandiver’s then certainly, it should be published.Thus Hasse decided to accept Grün’s manuscript for Crelle’s Journal.

However, in the form as presented so far Grün’s manuscript seemed not publish-able. Hence Hasse would first do what he always used to do as an editor of Crelle’sJournal: He would study the paper carefully and on that basis give advice to the authorto produce a text which, in his opinion, meets the standards of scientific publication.12

But he needed some time for this. Grün replied in a letter of December 19, 1932:

Vielen Dank für Ihr freundliches Schreiben von 12. Ich bin Ihnen sehrverpflichtet, wenn Sie sich dem Beweis zum Fermatproblem weiter wid-men wollen und es ist selbstverständlich, daß Sie jede Frist dazu haben.

Thank you very much for your kind letter of 12.13 I would be veryobliged to you if you would continue to attend to my proof on Fermat’sproblem, and it is clear that there will be no time limit for this.

We should note that just in this period, the last months of 1932 and the first monthsof 1933, Hasse was busy with his attempts to prove the Riemann hypothesis for curves.We have reported in [Roq04] that in November 1932, when Hasse gave a colloquiumlecture in Hamburg, he had a conversation with Artin who pointed out to him thathis (Hasse’s) research project on diophantine congruences was in fact equivalent tothe proof of the Riemann hypothesis for the curves in question. This comment byArtin had decidedly changed the viewpoint of Hasse. He went to work intensively onthis idea with the result that already in March, 1933 he arrived at his first proof forelliptic curves. In view of this we can understand that Hasse in this period tended topostpone, if possible at all, other obligations including the reading and correcting ofGrün’s manuscript. It was May 1933 until he turned to Grün’s manuscript again.

Grün’s paper [Grü34b] appeared in 1934. The date of submission is recorded asMay 17, 1933. His result in the final form reads as follows. As above, ` denotes anirregular prime number and � a primitive `-th root of unity. Let k0 D Q.� C ��1/.

12Rohrbach [Roh98] reports: “With his [Hasse’s] characteristic conscientiousness, he meticulously read andchecked the manuscripts…word by word and formula by formula. Thus he very often was able to give all kindsof suggestions for improvements to the authors, concerning contents as well as form.” The correspondenceHasse–Grün gives ample witness of this.

13This means December 19, 1932.


If the second class number factor h2 is prime to ` and if none of theBernoulli numbers B`n 0 mod `3 (for n D 2; 4; : : : ; ` � 3) then theequation

".� � ��1/m˛`1 C ˛`

2 C ˛`3 D 0

is not solvable in algebraic integers ˛1; ˛2; ˛3 2 k0 which are prime to`, provided m � 3` � 1 and " is a unit in k0.

This was Grün’s first publication. Compared with the other existing literatureon FLT it cannot be rated as exceptional. Grün followed the known footsteps in thedirection which had been pointed out by Kummer in the mid 19th century and hisresult was close to that of Vandiver [Van29]. But we should keep in mind that FLThad not yet been proved generally at that time. Hence any partial result which pointstowards the validity of FLT was welcomed, even if the progress compared with formerresults seemed to be small.

However, if we consider that Grün had originally not been aware of Vandiver’spaper and that his result containedVandiver’s, then we have to rate Grün’s achievementas extraordinary – in particular if we remember that he had no formal mathematicaltraining and had reached his high status of expertise through self-education.

2.3 From FLT to finite groups (1933)

In a letter of December 6, 1932 Grün started to discuss other problems; these belongto general class field theory and are only indirectly connected with FLT. Here we willnot go into all details but restrict our discussion to the following two topics.

2.3.1 Divisibility of class numbers: Part 1

Grün wrote to Hasse:

… Ich möchte noch einen Satz beweisen, der vielleicht gelegentlichgebraucht werden kann: Wenn K den Körper k enthält und kein Zwi-schenkörper existiert, der über k Abelsch mit der Relativdiskriminante 1ist, so ist die absolute Klassenzahl vonK durch die absolute Klassenzahlvon k teilbar.

… I would like to prove yet another theorem which may be useful oc-casionally: IfK contains the field k and there is no proper intermediatefield which is abelian over k and of relative discriminant 1 then the classnumber of K is divisible by the class number of k.

This is quite interesting. We know that five years earlier Artin had observed thesame fact, and he had found it worthwhile to communicate it to Hasse. Let us citefrom Artin’s letter of July 26, 1927:


Nun etwas anderes, das mir grossen Spass bereitet hat und das ich gesternim Heckeseminar erzählte. Das Resultat scheint, so trivial der Beweisist, neu zu sein. Eine ganz kindische Vermutung jedes Anfängers istdoch diese: Ist k Unterkörper von K, so ist die Klassenzahl von k einTeiler der Klassenzahl vonK. Ich möchte zeigen, dass dies „fast“ immerrichtig ist, mehr noch:

Satz: Enthält K=k…keinen in bezug auf k Abelschen und gleichzeit-ig unverzweigten Zwischenkörper, so besitzt die Gruppe der absolutenIdealklassen von K eine Faktorgruppe isomorph mit der Gruppe derabsoluten Idealklassen in k.

Now something else which I had talked about yesterday in Hecke’sseminar with great fun. The result seems to be new in spite of thesimplicity of proof. A very childish expectation of every beginner is thefollowing: If k is a subfield ofK then the class number of k divides theclass number of K. Now I show that this is true “almost always”, andeven more:

Theorem: If Kjk…does not contain any intermediate field which isabelian and at the same time unramified then the class group of Kadmits a factor group isomorphic to the class group of k.

Artin proceeds in his letter to describe a proof which, as he had said, is quitesimple. After checking we found that Grün’s proof was the same as Artin’s. Theessential fact to be used in the proof is that, under the hypothesis of the theorem, theabsolute class field of k is linearly disjoint to K. It seems that Hasse in his reply toGrün had mentioned Artin, for Grün wrote in his next letter (December 19, 1932) thathe did not wish to claim priority.14

Thus again, on his way teaching himself algebraic number theory, Grün had foundfor himself a theorem which was familiar to the specialists, this time Artin. Note thatArtin had never published his proof.

But there had been a recent publication by Chevalley [Che31] containing thesame theorem. Certainly Hasse, who at that time was in close contact with Chevalley,knew about Chevalley’s paper, and perhaps he had pointed out that paper to Grün afterreceiving Grün’s letter. At first sight Chevalley’s proof looks somewhat different thanthat of Artin–Grün but at closer inspection we find that it is essentially the same15.

14“Ich wollte den Satz nicht als mein geistiges Eigentum angesehen wissen.”15The difference between Chevalley’s andArtin’s arguments can be described as follows: Let k0 be the absolute

class field of k. Artin uses only the fact that k0 is abelian and unramified over k, and that these properties arepreserved after extending the base field from k to K – which directly implies the result. Chevalley uses theVerschiebungssatz (shift theorem) of class field theory in order to describe Kk0 explicitly as class field over K.Thus he uses more machinery from class field theory than Artin–Grün. However, if one comes to think of it, theproof of the Verschiebungssatz in this special case reduces to the argument of Artin–Grün and so, in this sense,we may regard both proofs as essentially the same.


Chevalley mentions that the same result would be contained in a forthcoming paperby Herbrand which we have found to be Théorème 2 in [Her32]; there Herbrand usedit for a new foundation of Kummer’s theory of ideal classes in cyclotomic fields. Afterchecking we found that again, Herbrand’s proof is the same as Artin’s and Grün’s.

From all we know about Grün we have no doubt that he had found his proofindependently, i.e., independent not only of Artin but also of Chevalley and of Her-brand. Grün in his letter cites Hilbert who in his Zahlbericht [Hil97], § 117, p. 37816,mentions that Kummer had stated the above theorem for the subfieldsK of Q.�/ butthat Kummer’s proof was incorrect.17 Of course, Kummer’s theorem is an immediateconsequence of the general theorem of Artin–Grün since Q.�/ is purely ramified.

But in Kummer’s case, i.e., for subfields of Q.�/where � is a prime power root ofunity, the theorem had been proved much earlier by Furtwängler [Fur08]. Althoughin 1908 class field theory was not yet completed by the theorems of Takagi and Artin,enough was known to prove the divisibility of class numbers which Kummer hadconjectured. It seems that neither Artin nor Grün had been aware of Furtwängler’sproof. But Hasse did know it, for in Hasse’s diary we have found an entry datedOctober 10, 1925 with the title: The ideal class groups of relatively abelian fields.(Generalization of a theorem of Furtwängler.)18 There, Hasse proved the Artin–Grüntheorem in the special case whenKjk is abelian. Thereby he regardsK as class fieldover k, thus he used still more machinery from class field theory. As it turned out inthe proof of Artin–Grün, this is not necessary. Here again, as it is the case so often inMathematics, the generalization (omitting the assumption that Kjk is abelian) leadsto a simplified proof.19

2.3.2 Divisibility of class numbers: Part 2

In his letter from December 19, 1932 Grün mentions another problem concerning classnumbers. While his above mentioned result yields a lower bound of the class numberh of K (it is divisible by the class number of a subfield under certain conditions), henow claimed to have an upper bound for h (under certain conditions it divides theclass number of a subfield times a certain factor dependent on the structure of theGalois group). This time, however, he is not sure that his arguments are correct, andso he writes:

Aber ich gestehe Ihnen, verehrter Herr Professor: Ich traue meinen eige-nen Ergebnissen nicht; die Sätze sind mir zu überraschend. Ich kann aber,

16The page number refers to the original Zahlbericht whereas its copy in the Collected Papers of Hilbert hasdifferent pagination.

17The same reference to Hilbert’s Zahlbericht we have found in Artin’s letter to Hasse, cited above.18At the end of this entry Hasse later had added a reference to Artin’s letter of July 26, 1927 which we have

cited above.19By the way, the Artin–Grün theorem with the same proof appears in [ACH65]. There, Hasse cites the letter

of Artin and also his own diary entry of October 10, 1925.


soviel ich mich auch bemühe, den Fehler nicht finden. Und deshalb bitteich Sie, mir zu sagen, ob und wo in meiner Rechnung ein Fehler steckt.

But I admit, dear Herr Professor, that I do not trust my own results: thetheorems are very surprising to me. However I cannot find the erroralthough I have tried to. Therefore I am asking you to tell me whetherand where there is an error in my computations.

The situation is the following: Kjk is a Galois extension of number fields20. LetK 0 be the maximal subextension which is abelian over k. Grün proved:

Suppose that the class group of K is cyclic. Then the class number hof K divides the product of the class number h0 of K 0 with the relativedegree ŒK W K 0�.

Actually Grün wrote that he assumed the cyclicity of the class group of K “forsimplicity only”, and claimed that his proof could be extended to cover the case ofan arbitrary class group. However that is not the case. Hasse pointed out this fact toGrün, and we shall see below that this led to remarkable consequences.

In the case of a cyclic class group of K, Grün’s proof turned out to be correct.But it seems that Hasse was not sure whether this result was known already, sincehe proposed to put this theorem as a problem in the Jahresbericht of the DMV.21

At that time, the Jahresbericht provided a section where any member could state aproblem, and the incoming solutions were published in the next issue. Quite oftensuch problems were posed even if the author had already obtained a solution, but hewished to find out whether a solution, possibly simpler, was known already.

Grün consented and Hasse submitted the theorem (for cyclic class group of K)under the name of Grün as a problem, which appeared as no. 153 in volume 43 (1934)of the Jahresbericht. Promptly there were two solutions received, published in vol-ume 44, one of L. Holzer and the other of A. Scholz, both being renowned numbertheorists. It turned out that both solutions were essentially the same as Grün’s originalproof in the letter to Hasse, and were independent of class field theory.

The proof is short and straightforward: One has to use the fact that the automor-phism group of a cyclic group is abelian and, hence, the commutator groupG 0 of theGalois group G of Kjk acts trivially on the ideal class group of K. Consequently,if c is any divisor class of K then the norm NKjK0.c/ equals cŒKWK0� and therefore,since NKjK0.c/ is a divisor class of K 0, we have that cŒKWK0�h0

is the principal class.Hence the exponent of the class group ofK divides ŒK W K 0�h0. Since the class groupof K is assumed to be cyclic the contention follows.22

20Grün considered only the case k D Q.21DMV = Deutsche Mathematiker Vereinigung = German Mathematical Society.22I am indebted to Franz Lemmermeyer for the information that Yamamura had rediscovered and used this

theorem of Grün. See [Yam97], p. 421. Lemmermeyer himself has used (and proved) this theorem in [Lem97],Proposition 6, citing Grün.


But the result seems to be quite special because of the assumption that the classgroup of K is cyclic. Therefore Hasse proposed to Grün to investigate the generalcase, with class group of arbitrary structure. Clearly, whenever a subgroup G1 of Gcan be found which acts trivially on the class group ofK then a similar argument canbe applied to obtain an upper bound for the exponent of the class group of K (notnecessarily for the class number h itself), with K 0 being replaced by the fixed fieldK1 of G1.

2.3.3 Representations over finite fields

On April 19, 1933 Grün answered that his attempts to deal with non-cyclic classgroups had not been successful. However after some time, on December 5, 1933 hewrote:

Nach langer Zeit kann ich Ihnen heute wieder etwas berichten. Ichhabe mich mit gruppentheoretischen Untersuchungen beschäftigt … Ichknüpfe an an meine Aufgabe No. 153 im Jahresbericht. Als ich Ihnendamals das Resultat mitteilte, stellten Sie die Frage: „Wie lautet dasgenaue Analogon für allgemeine Abelsche Klassengruppen?“ Um diesesProblem handelt es sich hauptsächlich.

After a long time I am able again to report something to you. I have beenbusy with group theoretical questions … I refer to my problem no. 153in the Jahresbericht. When I had reported to you on that result, youasked: “What is the exact analogue for arbitrary abelian class groups?”The following is mainly concerned with this question.

And Grün continues with a description of his results. Let G be a finite groupwhich acts on an abelian group A of exponent p, a prime number. (We observe thatGrün discussed, as a first step, not arbitrary abelian class groups but only p-groupsof exponent p, i.e., vector spaces over Fp .) Let m be the rank of A. For any prime` ¤ p let m` denote the order of p mod `. Grün wrote that indeed he has foundgeneral statements about subgroups of G which act trivially on A. He proved:

If ` > mm`

then the commutator group of an Sylow `-group of G actstrivially on A. In other words: If ` > m

m`then the Sylow `-groups of the

automorphism group of A are abelian.

Hasse had Grün’s manuscript refereed by Magnus who at that time was alreadyconsidered to belong to the leading German mathematicians in the field of grouptheory.23 Hasse asked him whether Grün’s result has appeared already in the litera-ture. Magnus replied that he knew only one source, an American paper by Brahana

23Wilhelm Magnus in Frankfurt had received his doctorate 1931 with Max Dehn as his supervisor. Thecorrespondence between Hasse and Magnus is preserved; it had started in 1930 when Magnus submitted hisdissertation [Mag30] for publication to Hasse as an editor of Crelle’s Journal.


[Bra34], which dealt with similar problems. Brahana’s result appears as a specialcase of Grün’s. And he added (letter of December 16, 1933):

… Ich finde die Sache wirklich sehr hübsch, auch der von ihm angegebe-ne Beweis des schon von Brahana gefundenen Spezialfalls … scheint miretwas durchsichtiger zu sein als bei B., und wenn sich die Ergebnisseauf Klassenkörperprobleme anwenden lassen, wäre das ja besonderserfreulich.

… I regard the matter as quite nice. Also, Grün’s proof in the specialcase which had already been found by Brahana … seems to me to besomewhat more transparent than B.’s proof. And if Grün’s results canbe used in class field theory then this would be particularly nice.

Obviously Hasse had written to him that he expects Grün’s results to be applicable inclass field theory. In fact, as we have seen, Grün’s group theoretical problem arosefrom a question about class numbers.

Grün in his letter also mentions that in addition to the above result, he has de-termined the structure of all Sylow groups, not only those for large `, of the auto-morphism group of an abelian p-group A of exponent p. Moreover, all those resultsare valid for the automorphism group of any vector space of finite dimension over anarbitrary finite field of characteristic p.

In fact, this is the content of the paper which Hasse finally accepted for Crelle’sJournal, already in the same year [Grü34a].

We see that Grün’s main interest had by now shifted to group theory – in conse-quence of Hasse’s question. The application to class field theory, he writes, will begiven later. But he never did so. It seems that from now on group theory absorbedall his interest.

2.4 The two classic theorems of Grün

More than one year later, on March 30, 1935, Grün submitted to Crelle’s Journalanother manuscript on group theory. This has turned out to become a classic andmade his name widely known [Grü35].

There are two main parts of the paper. In the first part he gives a direct gener-alization of what we have discussed in the foregoing section (and what had alreadyappeared in Crelle’s Journal). Namely, he dropped the condition on theG-moduleA:

LetG be a finite group which acts on an abelian p-group A of arbitrary structure,not necessarily of exponent p. Let m denote the rank of A. Let ` be a prime ¤ p.Then:


If k is the smallest exponent with `k > mm`

then the k-th commutatorgroup of an Sylow `-group of G acts trivially on A.

The paper contains an even further generalization, namely for an arbitrary p-groupA, not necessarily abelian, on which G acts. Then one has to consider the ascendingcentral series of A, and in the condition for `k the numberm has to be defined as themaximal rank of the factor groups of that series.

2.4.1 The second theorem of Grün

But the main results of this paper are to be found in the second part where we findthe two famous “Theorems of Grün”.

Given a finite group G and a prime number p, the problem is to describe thestructure of its maximal abelian p-factor group G=G.p/. Here, G.p/ denotes thep-commutator group of G. This description turns out to be particularly simple forgroups which have a property which is called “p-normal”.24 This property is definedas follows: the center C of a Sylow p-group P of G coincides with the center of anyother Sylow p-group in which C is contained. Grün proves:

If G is p-normal then the maximal abelian p-factor group G=G.p/ isisomorphic to the maximal abelianp-factor groupNC=N

.p/C , whereNC

denotes the normalizer of the center C of a Sylow p-group P of G.

The idea behind this is that a Sylow p-group P , its center C and the normalizer NC

are in general much smaller and easier to handle than the whole groupG. Hence thistheorem yields a criterion for a groupG to have a non-trivial p-factor group, namely:this is the case if and only if NC has this property. Note that NC contains C as anabelian normal subgroup, thus we have the situation which Grün considered in thefirst section of this paper, and that result is applicable to NC acting on C .

If in particular the Sylow p-group P ofG is abelian and is contained in the centerof its normalizer then G is p-normal and it follows the isomorphism G=G.p/ � P

which is a classical theorem of Burnside, and was well known also to Grün. In thissense Grün’s theorem can be regarded as a generalization of Burnside’s theorem –and a far reaching generalization at that.

The above theorem is usually called the “second theorem of Grün” although inGrün’s paper it is proved first, whereas the “first theorem of Grün” is what Grün provesafterwards. The switch in the enumeration is probably due to Zassenhaus25 who in hisgroup theory text book [Zas37] included the two theorems of Grün and introducedthe enumeration used today. This makes sense since Grün’s second theorem (inZassenhaus’ enumeration) can be regarded as a corollary of his first theorem.

24This terminology had been proposed by Hasse (letter to Grün of May 28, 1935).25Hans Zassenhaus got his doctorate 1934 under the supervision of Artin. From 1934 to 1936 he worked at

the University of Rostock, and there he wrote his famous text book on group theory.


2.4.2 The first theorem of Grün

The “first theorem” of Grün gives a description ofG=G.p/ for an arbitrary finite groupG, not necessarily p-normal. This is somewhat more complicated than in the case ofa p-normal group. Namely:

For an arbitrary finite group G, its maximal abelian p-factor groupG=G.p/ is isomorphic to the following abelian factor group of its Sylowp-group P :

G=G.p/ � P=P ?;

where the normal subgroup P ? � P can be described as

P ? D .P \N 0P /

Y�2G

.P \ ��1P 0�/:

Note that here appears the normalizer NP of the whole Sylow p-group P in G (notonlyNC ). As usual, P 0 denotes the commutator group of P , and similarlyN 0

P is thecommutator group of NP .

Admittedly, this result looks somewhat complicated because of the definition ofP ?. Nonetheless it has turned out to be quite important in group theory, in as muchas it shows that the maximal abelian p-factor group of any group G can be found asan explicitly given factor group of the (usually much smaller) Sylow p-group P . Itskernel P ? depends very much on how the Sylow p-group P is embedded into thegroup G.

We have already said that Zassenhaus, who at that time was writing a textbookon group theory, immediately recognized the importance of Grün’s theorems anddecided to include them into his book [Zas37].26

While reading this paper of Grün one can observe that its style is quite differentfrom that of his other papers. The paper is well written, careful in the use of notations,and it contains several diagrams which nowadays are known as “Hasse diagrams”.The explanation of this is that the manuscript, in the form as published, had beenentirely written by Hasse himself.

2.4.3 Hasse and the transfer

We have already said that Hasse, being an editor of Crelle’s Journal, used to checkevery manuscript carefully before sending it to print. So he did also with Grün’smanuscripts, and in particular with the manuscript under discussion. After all, Grünas an amateur had no experience with writing a paper. The letters of the Hasse–Grün correspondence show that Hasse worked quite hard to put this paper into shape.

26Zassenhaus, in his paper [Zas35b] on finite near-fields, had already discussed certain results centered aroundthe classical Burnside theorem as mentioned above. This may explain Zassenhaus’ great interest in results of thekind of Grün’s theorems.


After an extended exchange of letters there were so many corrections, additions anddeletions that the original manuscript was hardly readable any more. Finally Hasse,seemingly somewhat exasperated, proposed that he himself will now compose a newmanuscript. To which Grün replied (letter of May 18, 1935):

Ihre Mitteilung, daß Sie ein neues Ms. herstellen wollen, hat mich zwareinerseits hoch erfreut, aber – darf ich denn das annehmen ? …Ich weißwirklich nicht, ob ich das zugeben darf. Wir müßten natürlich auch Ihretätige Mitarbeit ausdrücklich vermerken. In jedem Falle …: Wenn Sievon Ihrer eigenen Zeit etwas opfern wollen, müssen Sie die betr. Sacheschon für sehr wichtig halten. Das ist das beste Lob, das ich mir denkenkann.

On the one hand, I am very glad about your proposition that you willcompose a new manuscript but – could I accept this ? … Really, I do notknow whether I am allowed to give my consent. Of course, we wouldhave to state explicitly your extensive cooperation. In any case …: Ifyou will spend your own time on this then you must consider it veryimportant. This is the best praise from you which I can imagine.

Hasse replied on May 21, 1935:

Ich halte es wirklich für das Beste, wenn ich hier ein neues Manuskriptherstelle. Die Arbeit daran würde mir Freude machen und Sie brauchensich darüber keine Gedanken zu machen. Dies in der Arbeit selbst zuerwähnen, würde mir nicht zusagen. Sie mögen das so auffassen, dasses zu meinen Aufgaben als Herausgeber gehört, wenn man diese imweiteren Sinne auslegt.

Indeed, I believe it is the best solution if I will write a new manuscripthere. It will be a pleasure to me and you do not have to worry aboutit. But I would not like that this be mentioned in the paper. You mightregard it as belonging to my tasks as an editor, if one interprets them ina wider sense.

Certainly, Hasse regarded Grün’s results as important and this was one of hismotivations to help Grün to put it into a form which would be appreciated by themathematical public. But another reason which required a complete rewriting ofthe manuscript, was Hasse’s proposition that the transfer map (Verlagerung) shouldbe used as an adequate tool which provides the isomorphisms of Grün’s theorems.For, in his original version Grün had not used the transfer and not obtained thoseisomorphisms, but he was content with saying that if one of the two factor groups(which we now know to be isomorphic) is non-trivial then the other is non-trivial too.

The transfer VG!H fromG to a subgroupH is a homomorphism fromG into thefactor commutator groupH=H 0. It can be defined as the determinant of the canonical


monomial representation ofG moduloH with coefficients inH=H 0. It had first beenconstructed and used by Schur [Sch02]. Later in 1927 the transfer was re-discoveredby Artin and Schreier during their attempts to prove the conjectured principal idealtheorem of class field theory. We know this from Artin’s letter to Hasse of August 2,1927. See alsoArtin’s paper [Art29]. It seems that neitherArtin nor Hasse were awareof the old paper by Schur because they never mentioned it in their letters nor in theirpublications. Artin was able, by means of his general reciprocity law, to reformulatethe principal ideal theorem as a purely group theoretical statement concerning thetransfer.27 Hasse in his class field report II [Has30a], p. 170 introduced the name“Verlagerung” for this group theoretical map, which then was translated into Englishas “transfer”.

By 1935 the transfer map was a well established tool but apparently it was usedmainly in number theory in connection with the principal ideal theorem and relatedquestions. It seems that in abstract group theory it had not yet found many applications(except in Schur’s paper mentioned above). But this changed after Grün’s paper.

In Grün’s letter of May 18, 1935 we read:

Haben Sie vielen Dank für Ihre Briefe und die darin enthaltenen wert-vollen Anregungen. Der Gedanke, die Theorie der Verlagerung her-anzuziehen, ist außerordentlich glücklich. Ich hatte ja auch bei meinemBeweis von Satz 5 ähnliche Wege eingeschlagen, ohne aber diese Theo-rie wirklich zu benutzen. Die Verlagerungstheorie gestattet, in einfacherWeise die Sätze 4 und 5 voll zu beweisen. Für Satz 4 haben Sie dies jaschon liebenswürdiger Weise so weit durchgeführt, …

Many thanks for your letters and the valuable suggestions therein. Theidea to use the transfer theory is extraordinarily fortunate. In my proofof theorem 5 I had used similar methods but without really using thattheory. Transfer theory leads to simple complete proofs of theorems 4and 5. In case of theorem 4 you have already kindly done it so far, …

Grün proceeds to expound in detail the proofs which Hasse had indicated usingtransfer theory. And later in this letter he writes:

Natürlich muß aber [in der Arbeit ] in jedem Falle darauf hingewie-sen werden, daß die Anwendung der Theorie der Verlagerung auf IhreAnregung hin erfolgt ist und ich somit diese eleganten Beweise Ihnenverdanke.

Of course, it should be mentioned [in the paper] that the application oftransfer theory is due to your suggestion and that, hence, I owe theseelegant proofs to you.

27One year later, in 1928, Furtwängler [Fur29] succeeded to prove this group theoretical statement. Laterthere were simplifications of Furtwängler’s proof, one also by Magnus [Mag34], but the most significant one byIyanaga [Iya34]. (By the way, Iyanaga says in the introduction that the greater part of his paper is due to Artin.)


But Hasse replied:

… scheint es mir aus sachlichen Gründen notwendig, in einer Fussnotezu erwähnen, dass der Gedanke, die Verlagerung bei den Beweisen vonSätzen Burnside’scher Art zu benutzen, von Herrn Ernst Witt, Göttingen,stammt.

… I find it necessary to mention in a footnote that the idea to use thetransfer in the proofs of theorems like Burnside’s is due to Mr. ErnstWitt, Göttingen.

Whereupon Grün, in a footnote to his paper [Grü35], inserted the following text:

Den Gedanken, bei diesem Beweis die ursprünglich von mir verwende-ten monomialen Darstellungen durch die Verlagerung zu ersetzen, ver-danke ich einer Mitteilung von H. Hasse. Dieser wurde seinerseits geleit-et durch eine mündliche Mitteilung von E. Witt, wonach sich der klas-sische Beweis des Burnsideschen Satzes in ganz entsprechender Weiseeinfacher und durchsichtiger gestalten läßt.

The idea to replace the monomial representations (which I originallyused) by the transfer map, arose from a suggestion of H. Hasse. He hadbeen led by an oral communication of E. Witt who pointed out that theclassical proof of Burnside’s theorem can similarly be simplified.28

There is another footnote, after the statement of the “first theorem of Grün”, readingas follows:

Diesem Satz und seinem Beweis hat Herr Hasse die vorliegende Formgegeben. Ich habe mich ursprünglich darauf beschränkt, bei den gemach-ten Voraussetzungen eine zyklische p-Faktorgruppe nachzuweisen.

This theorem and its proof has been put into the present form byMr. Hasse. Originally I had been content with showing, under the as-sumptions as stated, the existence of a cyclic p-factor group.

By this Grün means a non-trivial cyclic factor group of P=P ? as a necessary andsufficient condition that G=G.p/ is non-trivial. Certainly, the idea to establish groupisomorphisms (when possible) instead of only considering the group orders, is part ofthe “Modern Algebra” which had been propagated by Emmy Noether and had found

28Burnside’s theorem (as explained in Section 2.4.1) can be found in his book [Bur11], §243. The computationsperformed there are indeed the same as computing the kernel and the image of the transfer map in the specialsituation at hand. However, Burnside does not mention (nor does he care) that this is a general procedure,referring to a generally defined map. Therefore, if it is said that the definition of the transfer map goes backto Burnside, such statement has to be interpreted with appropriate caution. It takes some insight to realize thatBurnside’s arguments indeed can be looked at as evaluating a homomorphic map. We do not know whether Witthad known Schur’s paper [Sch02] or whether he had observed this himself.


its expression in van der Waerden’s text book [vdW31]. Hasse explained this to Grünin his letter May 28, 1935 as follows:

Entgegen Ihren brieflichen Andeutungen sehe ich allerdings doch dasHauptgewicht Ihrer Sätze in der Herleitung von notwendigen und hin-reichenden Bedingungen für die isomorphe Übertragung von Unter-gruppen oder Faktorgruppen innerhalbP auf Faktorgruppen vonG, undnicht so sehr in der blossen Folgerung auf die Ordnungen dabei. Daherhabe ich in den Formulierungen immer nur die Isomorphiebehauptun-gen angeführt und meine, man kann es ruhig dem Leser überlassen,die daraus ohne weiteres ablesbaren Folgerungen für die Ordnungen zuziehen.

Contrary to your hints in your letters I regard the main point of yourtheorems to be the isomorphic transport of subgroups and factor groupswithinP to factor groups ofG, and not so much in the mere consequencefor the group orders. Therefore, I have formulated all the theorems asreferring to isomorphisms. In my opinion it can be left to the reader todraw from this the consequences concerning the group orders …

Finally on June 7, 1935, when the manuscript seemed to have acquired a formsatisfactory to both, Grün wrote:

Lieber Herr Professor Hasse ! Vielen Dank für die Übersendung desManuskriptes und Durchschlages. Jetzt ist doch wirklich etwas aus mein-er ursprünglichen Arbeit geworden. Ich gestehe Ihnen, daß ich erst nunwirkliche Freude an ‚meinem‘ Manuskript habe.

Dear Professor Hasse ! Many thanks for the manuscript and carbon copy.Really, now there has developed something out of my original paper. Ihave to admit that only now I have real pleasure with ‘my’manuscript …

But the correspondence about this continued and several points had still to be cleared.It took until August 13, 1935, after more than eighteen letters29 had been exchangedbetween Hasse and Grün concerning this manuscript, that finally Grün could sendthe corrected proof sheets to Hasse. The paper appeared in the same year 1935 inCrelles Journal [Grü35].

We have reported about this part of the Hasse–Grün correspondence in a somewhatgreater detail, since it does not seem to be widely known to what extent Hasse had ashare in Grün’s classic paper. The title of the paper is:

Beiträge zur Gruppentheorie I.

Contributions to group theory I.29This means that eighteen letters have been preserved, six of them by Hasse and twelve by Grün. Those

letters of Hasse which are preserved are written with typewriter, and Hasse had made carbon copies. Probablyanother six letters by Hasse were handwritten and, hence, not preserved.


In one of his letters Grün had announced that there will be a second and perhaps moreparts of such “contributions”. But the next he submitted in 1943 only (due to theproblems in war time it appeared in 1945; see [Grü45]). Later in the course of timeGrün produced 10 such “contributions”, the last appearing 1964 again in Crelle’sJournal, when Grün was 74.30

2.4.4 Grün, Wielandt, Thompson

Let us jump 4 years ahead to the Göttingen group theory conference in 1939.31 Thereon June 27, 1939, Wielandt32 delivered a talk with the title: “Sylow p-groups and p-factor groups”. This is precisely the topic of Grün’s classic paper [Grü35] which wejust have discussed. In fact, Wielandt presented (among other results) a far reachinggeneralization of Grün’s result. The main theorem of Wielandt ist somewhat involvedand we do not reproduce it here. One of its many consequences concerns the casewhen a Sylow p-group P ofG is p-regular in the sense of Ph. Hall. This means that

xpyp .xy/p mod hx; yi0 p

holds for every x; y 2 P . (In other words: The operation “p-th power” can beperformed termwise, modulo a product of p-th powers of commutators from thegroup generated by x and y.) Under this assumption it follows from Wielandt’s mainresults that the maximal p-factor group of G is isomorphic to the maximal p-factorgroup of the normalizer NP . Note that here the p-factor groups in question may benon-abelian whereas Grün’s results refer to abelian p-factor groups only. Wielandtachieves this by manipulating the monomial representation directly in a suitable way,not only the transfer map which is the determinant of the monomial representation.

Wielandt’s talk was published 1940 in [Wie40]. It is evident that Wielandt’spaper is directly influenced by Grün’s. B. Huppert has given following comment toWielandt’s paper:33

Eines der Ziele von Wielandt wird in dieser Arbeit mit keinem Worterwähnt, nämlich die Nilpotenz des Frobenius-Kerns einer Frobenius-Gruppe. Diese wurde zuerst von J. Thompson bewiesen. Im Sommer1958 gab es in Tübingen eine lange Unterhaltung zwischen Wielandtund Thompson. Unmittelbar danach sagte Wielandt zu mir: „Das ist einsehr scharfsinniger Bursche, von dem kann man etwas lernen.“ EinigeMonate später reichte Thompson seine Arbeit bei der MathematischenZeitschrift zur Publikation ein. Demnach gibt es eine ganz deutliche

30In Grün’s enumeration there were “Contributions” no. I–IX and XI published, but not no. X. We do not knowhis plans for no. X.

31For more on this conference see Section 2.6.2.32Helmut Wielandt had studied in Berlin with I. Schur and was awarded his doctorate in 1935. In 1939, the

year of the Göttingen group conference, he held a position of assistant at the University of Tübingen.33Private communication.


mathematische Verbindungslinie von Grün über Wielandt bis zu Thomp-son.

One of Wielandt’s motivations is not mentioned at all in this paper[Wie40], namely to prove the nilpotency of the Frobenius kernel of aFrobenius group. This was proved later only by J. Thompson [Tho59].In the summer of 1958 there was a long discussion in Tübingen betweenWielandt and Thompson. Immediately thereafter Wielandt said to me:“This is a very sharp-witted guy, from him one could learn a lot.” Sev-eral months later Thompson submitted his paper [Tho59] to Wielandtfor publication in the Mathematische Zeitschrift. Thus we can observevery clearly a line of mathematical influence from Grün over Wielandtto Thompson.

Remark. R. W. van der Waall has pointed out to me that the line of mathematicaldevelopment which started with Grün’s paper can be traced much further: There arequite a number of subsequent papers continuing the ideas of Grün and supplementinghis results. Of particular interest is the following result contained in a paper byT.Yoshida published in the Journal of Algebra 52 (1978), pp. 1–38. It says that thetransfer isomorphism G=G.p/ � NP =N

.p/P holds quite generally, with exceptions

possible only if P admits a factor group isomorphic to the wreath product of thecyclic group of order p with itself. Indeed this is a very strong generalization of thefirst theorem of Grün. The transfer map and its dual have become standard tools inthe theory of finite groups.

2.5 Grün meets Hasse (1935)

2.5.1 Hasse’s questions

Grün, in his first letter to Hasse, had introduced himself as an amateur mathematician.But it seems that Hasse, impressed by Grün’s achievements, had some doubts by now.Although there had been an exchange of letters since three years, he did not knowanything definite about Grün’s mathematical background. So Hasse at last asked inhis letter of May 8, 1935:

… Sind Sie eigentlich Mathematiker von Hauptberuf, oder treiben Siedie Mathematik nur nebenbei als Liebhaberei ?

… By the way, are you a mathematician by profession, or are you doingMathematics as a hobby?

To which Grün replied (letter of May 9):


Ich wollte, verehrter Herr Professor, ich wäre Mathematiker von Haupt-beruf. Leider ist das nicht der Fall, ich muß mich ohne besondere Begei-sterung kaufmännisch betätigen, um zu leben.

I would wish, dear Herr Professor, that I could be a professional mathe-matician. Unfortunately this is not the case; I have to work for a livingin a commercial job, though without particular enthusiasm.

But Hasse continued to inquire (letter of May 13):

Wo haben Sie sich denn Ihre mathematischen Kenntnisse erworben?Haben Sie einen bestimmten Mathematiker zum „Lehrer“ gehabt?

But where did you pick up your mathematical knowledge? Have youhad a “teacher” who was a mathematician?

Grün’s reply (letter of May 15):

Ob ich einen bestimmten „Lehrer“ gehabt habe ? Ich habe meine Kennt-nisse nur aus Büchern geschöpft und da sind Sie selbst zu einem großenTeil mein Lehrer gewesen. Ich bekam zufällig Ihre beiden Berichte indie Hand und damit begann mein intensives Interesse für Klassenkör-pertheorie. Natürlich war ich mathematisch so weit vorgebildet, daß ichfähig war, die Berichte durchzuarbeiten. Die außerordentliche Klarheitund Durchsichtigkeit Ihrer Darstellung nimmt ja dem Leser jede Ar-beit ab. Bis dahin hatte ich mich eigentlich mehr für Funktionentheo-rie interessiert, allerdings hatte ich wenigstens Hilberts „Zahlbericht“,Dirichlet, Dedekind und die einzelnen Kummerschen Arbeiten gelesen.Nun wurden Ihre Berichte für mich Veranlassung, mich intensiv mitGruppentheorie zu befassen.

Whether I have been taught by a particular teacher? I have acquired myknowledge from books only, and there to a large degree my teacher hasbeen you. Your two reports34 came by chance into my hands, and thisstarted my intensive interest in class field theory. Of course I had alreadyacquired enough of the mathematical prerequisites which enabled me toread your reports. After all, the wonderful clarity and transparency ofyour presentation spares the reader much of the work. Until then I tendedto have more interest in the theory of complex functions, but I had alreadyread Hilbert’s “Zahlbericht”, Dirichlet, Dedekind and various papers byKummer.35 Now your reports had induced me to look intensively intogroup theory.

34Grün refers to Hasse’s class field reports, the first on Takagi’s class field theory [Has26a], and the second onArtin’s reciprocity law [Has30a].

35At that time, the “Collected Papers” of Dirichlet and Dedekind were available, but not yet Kummer’s. Thelatter would be published in 1975 only, edited by André Weil.


When Grün states that his interest in group theory had been induced by Hasse’sclass field report, then we see that by now he had well grasped the main trend inthe then “modern” class field theory, as expressed in the foreword to Part II of thatreport:36

Artin’s Reciprocity Law constitutes an advance of the utmost impor-tance. Its importance lies not so much in the direction which might besuggested by the name “reciprocity law” and its classical formulation,but in the general class field theory. The ultimate aim of it is the cod-ing of all arithmetical properties of a relative abelian number field in itsGalois group, similarly as the aim of Galois theory is the coding of fieldtheoretic properties in the Galois group.

However, Grün’s mathematical interests had now shifted from FLT and class fieldtheory almost entirely to group theory. The application to class field theory does notappear in his further publications. In group theory Grün had found his main subjectwhere he would be active in the future. A majority of 21 of his total of 26 papersfrom 1934 to 1964 belong to group theory.

2.5.2 Grün’s visit

In view of this correspondence, Hasse now wished to meet Grün personally, in par-ticular since Grün had announced to have many more results in his files. For, in hisletter of May 9, 1935 Grün had written:

Nach der Veröffentlichung meiner beiden Noten über den Fermat undGruppen im Galoisfeld hat mir das Kultusministerium eine gewisse Un-terstützung zuteil werden lassen, die mich in Stand setzte, mich einigeZeit fast ausschließlich mathematischen Untersuchungen zu widmen.Die Folge ist, daß ich geradezu eine Unmenge von Notizen habe, in de-nen die wesentliche Vorarbeit für eine Veröffentlichung schon geleistetist; alle diese Arbeiten sind gruppentheoretischer Natur, natürlich mitkörpertheoretischen Anwendungen.

After publication of my two notes on Fermat and on groups in a Galoisfield the ministry of education had granted me a certain stipend whichenabled me to devote almost all my time to mathematical work. Asa consequence I have a huge pile of notes which already contain theessential ingredients of future publications. All of this work is of grouptheoretical nature, of course with applications to field theory.37

36The following is a free translation of essential features of Hasse’s foreword of [Has30a]. – The reader maycompare this with Hasse’s foreword in his book on abelian fields [Has52].

37To this Grün added: “I have to acknowledge with thanks the support which I have found with the ministerof education, for neither was I a member of the party nor have I become such.” Of course, the “party” which


So Hasse wrote on May 13, 1935:

Wir haben hier in diesem Semester gerade eine kleine Arbeitsgemein-schaft über Gruppentheorie, in der wir mit Ihren Untersuchungen sehrverwandte Dinge betreiben, insbesondere die beiden neuen gruppentheo-retischen Arbeiten von Zassenhaus studieren, die im letzten Heft der Ab-handlungen des Hamburger Mathematischen Seminars erschienen sind.

In this semester we have here a small workshop on group theory, ontopics which are closely related to your investigations. In particular weare studying the two new group theoretic papers of Zassenhaus whichhave appeared in the last issue of the Hamburger Abhandlungen.38

And Hasse continued:

Sehr gerne würde ich Sie auffordern, doch im Monat Juni einmal hier-her zu kommen und bei uns in der Arbeitsgemeinschaft über Ihre grup-pentheoretischen Studien vorzutragen, ganz zwanglos, d. h. so dass mandazwischenfragen darf, wenn man etwas nicht versteht, und das ganzemehr den Charakter einer gemeinsamen Erarbeitung hat.

I would like very much to invite you to visit us some time in June, and toinform us about your group theoretic work. This should be completelyinformal, so that it will be possible to put questions; the whole thingshould have the character of a common discussion.

On June 13, 1935 Grün arrived in Göttingen39; his talk in the workshop was sched-uled for the next day, a Friday. Hasse had offered him to lodge in the MathematicalInstitute where there was a visitor’s room available, and to stay over the weekend inorder to have opportunity for discussions with the people of, in Hasse’s words, “thesmall but lively group of algebraists” in Göttingen. We know from other sources thenames of the members of that group, the most outstanding members besides Hassebeing Witt, Teichmüller and also H. L. Schmid.40 The latter was to play, ten yearslater, an important role in Grün’s life.

he alludes to, was the NSDAP, the Nazi party which had come to power in Germany in January 1933. Indeedit seems remarkable that Grün was supported in his work by the government of that time although he did notconform to the official party line. Later in 1946 he wrote that he had to suffer severe personal repression becausehe repeatedly had been urged to join the party but always refused.

38These were the papers [Zas35a] and [Zas35b], the first one on the characterization of linear groups aspermutation groups, and the second on finite near-fields.

39Note that in the summer of 1934 Hasse had left Marburg and accepted a position at the University ofGöttingen. Thus Hasse’s invitation to Grün was meant for Göttingen, not Marburg. For details of Hasse’s changeto Göttingen in the midst of the political upheavals of the time, we refer to [Fre85] and [Sch87].

40This was the same Arbeitsgemeinschaft in which one year later the Witt vectors were discovered, togetherwith their application to cyclic extensions in characteristic p and class field theory, as well as to the structuretheory of p-adic fields. Those results are all published in one volume of Crelle’s Journal (vol. 176), together withthe seminal paper of Hasse who used Witt vectors for the explicit p-power reciprocity law of class field theory.


In a former letter Grün had asked whether his talk in the workshop could be aboutp-groups, and Hasse had replied that the choice was entirely up to the speaker. And,knowing from his correspondence that Grün may have some problems to explainmathematical arguments in a correct form, Hasse had added the advice that Grün inhis talk should be very explicit in all details.

Perhaps it is not without interest to cite Hasse’s words where he tried to informGrün about what had been discussed in the workshop so far, i.e., what he could assumeto be known:

Über p-Gruppen haben wir auch schon gesprochen. Wir haben die klas-sische Theorie (Speiser) durchgenommen, ferner noch einige weitereSätze über die Anzahlen der Untergruppen oder Normalteiler gegeben-er Ordnung in einer p-Gruppe. Weiter die Theorie der HamiltonschenGruppen (alle Untergruppen Normalteiler ) und der p-Gruppen, in de-nen es nur eine Untergruppe der Ordnung p gibt (nur für p D 2 gibtes nicht zyklische solche Gruppen). Ich werde morgen über Satz 5 undSatz 9 Ihrer Arbeit vortragen.

We have already discussed p-groups. We worked through the classicaltheory (Speiser)41 and in addition some theorems about the number ofsubgroups and normal subgroups of given order in a p-group. Further-more the theory of Hamiltonian groups (all subgroups are normal), andthe p-groups with only one subgroup of order p (only for p D 2 thereare non-cyclic groups with this property). Tomorrow I shall talk abouttheorems 5 and 9 of your paper.

Theorems 5 and 9 were the second and the first theorem of Grün as discussed above.The above lines show that in the circle around Hasse there was lively interest to

learn more about the newest results of finite groups, in particular p-groups. Thismay have its explanation by the fact that during those years the theory of p-groupshad been used heavily in algebraic number theory. We only mention the work ofArnold Scholz (who has had an extensive exchange of letters with Hasse) and whojust recently had proved the existence of number fields with a given p-group of classtwo as Galois group [Sch35b]. (And one year later Scholz would prove the same foran arbitrary finite p-group [Sch37].) This gives us perhaps another clue why Hassewas so much interested in Grün’s results on p-groups.

Unfortunately we have not found any record about what Grün had actually talkedabout, nor how his talk was received by his young audience. Did Grün indeed talkabout p-groups and what were his results which he presented? We can imagine thatGrün, not being used to lectures and colloquium talks, had some difficulties to addresssuch a group of brilliant young mathematicians who were used to high standards notonly with respect to the mathematical topics under discussion but also as to the way

41Hasse means Speiser’s monograph on group theory [Spe27].


of presenting new material. Doubtless Grün met high respect among these people, inview of his outstanding results so far. But did they appreciate his talk? From othersources (in later years) we infer that Grün’s talks used to be somewhat clumsy anddifficult to follow.

One week after Grün’s talk, Zassenhaus visited the workshop in Göttingen, onJune 21, 1935. Hasse had offered Grün to stay longer in order to meet Zassenhaus,and Grün did so. Note that Grün’s paper [Grü35] had not yet appeared, and thatZassenhaus was just working on the text of his group theory book [Zas37]. It seemsprobable that Zassenhaus, when he met Grün in Göttingen, learned about Grün’stheorems and realized their importance. In the foreword to his book (which appearedin 1937) Zassenhaus says that he wished to include the new and far-reaching resultsin group theory of the last 15 years; certainly Grün’s theorems were among those andthus found their way into Zassenhaus’ book.42

Two months after Grün’s visit to Göttingen he wrote to Hasse (letter of August 13,1935):

Lassen Sie mich Ihnen nochmals danken für die Gastfreundschaft, dieich in Göttingen gefunden habe. Es war geradezu eine Wohltat für mich,einmal nur mit wissenschaftlichen Problemen beschäftigt zu sein. Wennnicht meine wirtschaftliche Lage etwas anderes forderte, würde ich michin Göttingen niederlassen und mich völlig meinen mathematischen Un-tersuchungen widmen.

Thank you again for the hospitality which I have found in Göttingen. Itwas really a great pleasure to me to be occupied exclusively by scientificproblems. If my economic situation would have been different then Iwould settle in Göttingen and would occupy myself completely withmathematical research.

This sounds as if Grün had hoped to be offered a position at the University of Göttingenwhich would enable him to exclusively follow his research work. But this was notthe case.

With the same letter Grün returned the proof sheets of his paper [Grü35]. Recallthat the title of that paper carried the label “Part I” which implied that there wouldbe more parts, at least a second part. Accordingly, Grün mentioned in his letter hisplans for “Part II”, and that this would include investigations on p-groups. From thiswe may perhaps conclude that indeed, his talk in Göttingen was about p-groups, andthat he had been asked to send a manuscript about his talk to Crelle’s Journal, to bepublished as Part II of his “investigations”.

42Zassenhaus book on group theory has been said to have been “for decades the bible of the group theorists”(Reinhold Baer). – Nowadays both Grün’s theorems do appear in many textbooks on group theory, for instancein Huppert [Hup67]. Perhaps it is not without interest to note that Grün’s theorems have been included andgeneralized in the setting of homological algebra. See, e.g., the book of Cartan–Eilenberg [CE56] chap. XIItheorem 10.1.


But this Part II did not materialize in the form as planned. Several months later,in a letter of February 7, 1936, Grün apologized to Hasse that the envisaged paper onp-groups is not yet finished. He announced the manuscript to be finished in abouttwo weeks, but finally it took several years for this. And the real Part II, which wehave said appeared in 1945 only, did not deal particularly with p-groups [Grü45].

2.6 The Burnside problem (1939)

2.6.1 Dimension groups

After the appearance of Grün’s paper [Grü35], his exchange of letters with Hasseslowed down in frequency and intensity. Grün had found his main interest to begroup theory. He knew that Hasse’s main interest was number theory, and so he mayhave felt that now he could pursue his work without having to rely every time onHasse’s advice.43

In the year 1936 there appeared the paper [Grü36] on the descending central seriesof free groups. This paper is never mentioned in the Hasse–Grün correspondence.Grün proves, with an unusual and somewhat peculiar argument using group repre-sentations, that the “dimension groups” as defined by Magnus [Mag35] do coincidewith the members of the descending central series of the given free group. This wasconsidered an important result.

Since Grün’s paper directly refers to a paper by Magnus it is not unreasonable toassume that Grün had discussed it with Magnus before publication. Maybe it wasMagnus himself who had posed the problem to Grün. We know from several sourcesthat there was mathematical contact between Grün and Magnus in those years since1935. But the correspondence Grün–Magnus seems to be lost and so we do not knowthe details of how strong Magnus’ influence had been for this paper.

In any case, one year later Magnus himself provided a simplified proof, publishedin Crelle’s Journal [Mag37]. But Grün’s proof was duly registered as the first, andwas appreciated by the specialists.

2.6.2 The group theory conference in Göttingen

In June 1939 Hasse had organized a 5-day group theory conference in Göttingen.About the preparations for this conference we read in a letter which Hasse had

sent jointly to Magnus and Zassenhaus, dated February 18, 1939:

Die Göttinger Mathematische Gesellschaft plant in der letzten Wochedes Sommersemesters 1939 eine grössere Vortragsveranstaltung über

43In [JL98] it is said that, according to Grün himself, it was Hasse who had advised him to switch from numbertheory to group theory.


das Thema Gruppentheorie. Wir haben dazu Herrn P. Hall von King’sCollege, Cambridge eingeladen, uns drei grössere Vorträge aus seinemArbeitsgebiet zu halten. Zu meiner grossen Freude hat Herr Hall sichdazu bereit erklärt …

The Mathematical Society of Göttingen is planning a conference on“Group Theory”. We have invited Mr. Ph. Hall from King’s college,Cambridge, for three lectures from his field of research. I am very gladthat he has consented …44

Hasse then explained that the lectures of Philip Hall should form the core of theconference, but in addition he wished that a number of German mathematicians whowere working in group theory, should be given the opportunity to participate as invitedspeakers. And Hasse asked Magnus and Zassenhaus to help him with their expertiseand advice to prepare this conference.

In the ensuing correspondence between Hasse, Magnus and Zassenhaus it wasdecided that not too many talks should be scheduled, which meant that only thoseGerman mathematicians should be invited as speakers whose field of research hadsome connection to Hall’s, which is to say mainly p-groups and solvable groups andrelated topics. This then would include Grün, as Hasse observed:

Wenn Grün gewonnen werden könnte, so wäre das natürlich sehr schön.Er hat doch bei allem Ungeschick seiner Darstellung die Gruppentheorieum einige wichtige Erkenntnisse bereichert, die in engstem Zusammen-hang mit den Hallschen Arbeiten stehen. Ich bitte Herrn Magnus, sichmit ihm in Verbindung zu setzen.

If Grün could be won over then this would be very nice indeed. Notwith-standing his awkwardness in the presentation of material, he has enrichedgroup theory with some important discoveries which are very closelyconnected with Hall’s papers. I am asking Mr. Magnus to get in touchwith him.

When Hasse mentioned the “awkwardness in the presentation” then he may haverecalled his experiences four years ago with Grün’s paper which he (Hasse) had torewrite completely. Maybe Grün’s talk in the Göttingen Arbeitsgemeinschaft hadalso added to this impression. Nevertheless, in view of Grün’s achievements Hassedid not hesitate to name him as invited speaker of the conference.

And when Hasse asked Magnus to get in touch with Grün, then this reflects thefact that, as said above, by now the mathematical contact of Grün with Magnus hadbecome closer than his contact with Hasse.

44In the end, Hall delivered four lectures. – Hall was criticised for going to Germany at this difficult time buthe defended his actions saying: “ … the German mathematicians … [are] as little responsible for the presentsituation (and probably enjoy it as little) as you or I do.” (Cited from The MacTutor History of Mathematicsarchive.)


The Göttingen group theory conference took place from June 26 to June 30, 1939.The program is published in 1940 in volume 182 of Crelle’s Journal, together withthe papers presented at the conference.45 Hence it will not be necessary to go intoall details here. The paper of Grün [Grü40] has the title:

Zusammenhang zwischen Potenzbildung und Kommutatorbildung.

The connection between forming powers and commutators.

The paper is motivated by and closely connected to the old

Burnside problem. Is every finitely generated group of finite exponentnecessarily finite ?

See [Bur02]. Form D 2 the problem has a positive answer, already given by Burnside.This is so because every group of exponent 2 is commutative, as a consequence ofthe formula

t�1s�1ts D t�2.ts�1t�1/2.ts/2

which expresses commutators as products of squares. This led Grün in his paper tostudy similar formulas connecting commutators and powers.

The Burnside problem has also a

Restricted version. Are there only finitely many finite groups with agiven number r of generators and a given exponent m?

Grün’s paper [Grü40] was the first in which this “restricted” Burnside problem wasspecifically addressed, but not under that name. The term “restricted Burnside prob-lem” was coined later by Magnus [Mag50].

Let Fr denote the free group with r generators, and Fmr the subgroup generated

by the m-th powers. The Burnside problem asks whether the factor group Fr=Fmr is

finite. In his paper Grün considers the case whenm is a prime power pk; this impliesthat the group of Fr=F

mr and its factor groups are p-groups.

Grün observes that the restricted Burnside problem has an affirmative answerfor the pair r , m if and only if the descending central series of Fr=F

mr terminates

after finitely many steps. Note that the descending central series is defined by com-mutators, and so the above condition requires certain relations between powers andcommutators. In his proof Grün used his results of his former paper [Grü36], as wellas results of Magnus [Mag35], of Witt [Wit37b] and Zassenhaus [Zas39].

Grün’s paper was refereed in Zentralblatt by Zassenhaus, in Fortschritte derMath-ematik by Speiser, and in the newly founded Mathematical Reviews by Baer. In thereview by Zassenhaus we find the statement that Grün solved the restricted Burnsideproblem in the positive sense for r D 2; m D 5. Baer in his review says “the authormay prove that …” without saying that he really had proved it. The computations

45Among them the paper by Wielandt which we have mentioned in Section 2.4.4.


are quite involved and it seems that nobody had checked it. Later Kostrikin [Kos55]claimed that he had proved the restricted Burnside problem for r D 2, m D 5 butagain, this seemed to be doubtful until Higman [Hig56] independently had settled thequestion positively, for aritrary r and m D 5. 46

Although Grün’s paper carries the date of receipt as of August 21, 1939, Hassewould accept it only after it had been checked carefully by Magnus. With Magnus’help the paper underwent a thorough clean up. On January 21, 1940 Magnus wrotefrom Berlin47 that he had worked the last two weekends with Grün, and that the latterhad promised to complete his manuscript until the next weekend. The final versionready for printing arrived at Hasse’s office on January 31, 1940.

The attentive reader will have observed that between the dates involved, June 1939(date of the Göttingen conference) and January 1940 (receipt of Grün’s paper in finalversion) there was September 1, 1939, the outbreak of World War II. The publicationof the conference papers in Crelle’s Journal was somewhat delayed because one ofthe authors, Philip Hall, was a citizen of a country which now was in state of war withGermany. Hence it was necessary for Hasse to obtain the permission of the properGerman governmental offices to publish Hall’s papers in Crelle’s Journal. When thatpermission was finally granted it turned out that only two of the four anticipatedpapers by Philip Hall had arrived. Since postal service between Germany and GreatBritain had ceased there was no hope that the two missing articles would arrive byordinary mail, and Hasse had to find other ways to obtain those articles. This wasfinally possible with the good services of Carleman at Djursholm who resided inSweden, a neutral country.

2.6.3 A letter of 1952

Although Grün’s power-commutator formulae in [Grü40] turned out to be useful inseveral respects, they did not lead Grün to the general solution of Burnside’s problem,restricted or not, as he had hoped.

But Grün did not give up. Twelve years later, on June 30, 1952, after Hasse hadsent him gratulations for his 64th birthday he thanked Hasse for it and then wrote:

… ich habe ein Ergebnis erhalten, das ich sehr hoch einschätze: Die ab-steigende Zentralreihe hat gesiegt! Die Vermutung von Burnside „Setztman in einer aus endlich vielen Elementen erzeugten freien Gruppe F

46For arbitrary parameters r , m the restricted Burnside problem has been finally solved in the positive senseby E. Zelmanov who had been awarded the Fields Medal in 1998.

47Magnus was in Berlin at that time. From the correspondence Hasse-Magnus we know that one year earlier,at the annual DMV-meeting in Baden-Baden, he had approached Hasse and asked whether Hasse could helphim to find a new job since his position of Privatdozent at the University of Frankfurt had become unsustainablefor political reasons. Hasse was able, with the help of Wilhelm Süss who had acquired some influence in theministry of education, to find for Magnus a position at the University of Königsberg. Magnus went there for thesummer semester 1939 but then accepted a job in industry with the electronic company Telefunken in Berlin.


allem-ten Potenzen gleich 1,m eine beliebige natürliche Zahl, so entste-ht eine endliche Gruppe“ ist irrig. Es gilt im Gegenteil: … Fr=F

mr ,

kann nur dann endlich sein, wenn entweder Fr zyklisch .r D 1/ oderm D 2i3k ist. In allen anderen Fällen ist Fr=F

mr gewiß unendlich.

I have obtained a result which I estimate quite highly: The descendingcentral series has won! The conjecture of Burnside, “If in a finitelygenerated free group F allm-th powers are put to 1 then there appears afinite group”, is not true. On the contrary: … Fr=F

mr can be finite only

if either Fr is cyclic (r D 1) or m D 2i3k . In all other cases Fr=Fmr is

infinite.

Hasse replied on July 15, 1952:

Was Ihr neues Resultat betrifft, so ist das ja in der Tat ganz aufregend.Herr Witt, dem ich sofort davon Mitteilung machte, meinte, Sie hättenwohl das Resultat nicht ganz präzis mitgeteilt, denn bei zwei Erzeugen-den sei doch im Falle m D 5 bekannt, dass die Gruppe endlich sei.

Concerning your new result, this is indeed very exciting. I have imme-diately informed Mr. Witt48, and he thinks that you had not stated theresult in sufficiently precise form, for with two generators andm D 5 itis known that the group is finite.49

And Hasse asked Grün to send him the precise formulation of the result.We do not know Grün’s proof but since he did not reply to Hasse and did not

publish this result there was probably an error in it. Maybe Grün had shown his proofto Magnus who pointed out the error. Note that Magnus had published two yearsearlier another paper connected with Burnside’s problem [Mag50], hence he was stillinterested and informed about the problem.

At the DMV-meeting 1953 in Mainz, Grün had announced a talk mentioning theBurnside problem and the Baker–Hausdorff formula in the title. In the same yearGrün published a paper [Grü53] on p-groups in the Osaka Mathematical Journal inwhich some connections to the Burnside problem were given. The paper was rated asan “interesting paper” by Suzuki in his Zentralblatt review. But apparently nothingdecisive concerning the Burnside problem came out of these activities.

So this is another case where Grün had attempted to solve a famous great problembut failed in the end, although he was able to contribute interesting methods, formulasand lemmas.

48In 1952 Hasse and Witt were colleagues at the university of Hamburg.49I am somewhat puzzled by Witt’s statement. As far as I know the Burnside problem in the unrestricted sense

is still open in the case r D 2 and m D 5. Did Witt have a proof which he never published? Or did Witt refer tothe restricted Burnside problem? But the text of Grün’s letter indicates that he is concerned with the unrestrictedproblem.


2.7 Later years (after 1945)

Grün had expressed in one of his first letters to Hasse that he did not particularlylike his commercial job, and that he wished to be free to do mathematical researchexclusively. In the year 1938 he finally had the opportunity to leave his unbelovedcommercial job (whatever it was). As he reports in his vita50:

Auf Bemühungen einflussreicher Mathematiker wurde ich 1938 Chef-mathematiker am Geophysikalischen Institut in Potsdam.

Due to the help of influential mathematicians I was appointed chiefmathematician at the Geophysics Institute in Potsdam.51

But we are somewhat doubtful whether this new job did leave him much more timefor group theory research as did his former job. (Although, as we have seen inSection 2.5.2, he could participate in the Göttingen group theory conference in 1939.)In any case, during the war years until 1945, Grün was drafted to work as an “expert” atthe Navy Headquarters in Berlin52; from this work there resulted a paper on theoreticalphysics (which was published later in 1948 [Grü48b]). Again it does not seem likelythat in this period Grün had much time to spare for group theory.

After the war Grün found himself in the devastated city of Berlin without a job,hence free to tend exclusively to his mathematical research, but also without anyincome. In this situation he was picked up by Hermann Ludwig Schmid.

2.7.1 H. L. Schmid and Grün

The mathematical scene in Berlin of the immediate post-war years has been vividlypictured by Jehne and Lamprecht [JL98].53 H. L. Schmid was the main figure whotook the necessary initiative and started to rebuild Mathematics at Berlin Universityand at the Berlin Academy from level zero. He was successful to attract mathemati-cians of high standing to Berlin, like Hasse and Erhard Schmidt (and others). He builtand managed the new editorial office of the Zentralblatt der Mathematik in Berlin.Against many obstacles he founded a new mathematical journal, the MathematischeNachrichten, and served as its managing editor. Using his diplomatic skills he suc-ceeded to create a quiet atmosphere where mathematical life could prosper, protected

50We are referring to the same vita from which we have cited in Section 2.2.1.51I do not know the identity of the “influential mathematicians” mentioned by Grün. It seems unlikely that

it was Hasse; the topic of Grün’s job in Potsdam was never mentioned in their correspondence. – But see theappendix!

52“Sachverständiger beim Oberkommando der Marine”, according to his own words in his vita. – We do notknow whether it was the same military department where Hasse and a group of other mathematicians (includingMagnus) were working during the war years.

53Klaus Krickeberg has pointed out to me that the article [JL98] describes only part of the “mathematicalscene” in Berlin of those years. Another part was dominated by Erhard Schmidt in the direction of analysis.


from an evironment full of all kinds of basic day-to-day problems. He “led mathe-matics in Berlin to a first revival”.54 For a time it looked like Berlin could become aleading center in Germany for Mathematical Sciences.

H. L. Schmid took Grün under his wing and was able to get him some financialsupport, first in the University of Berlin55 and since 1947 in the newly foundedMathematics Research Institute of the Berlin Academy of Science.56

H. L. Schmid had been assistant to Hasse in 1935, and he had met Grün whenthe latter visited Göttingen (see Section 2.5.2). Since 1940 H. L. Schmid workedin Berlin as an assistant to Geppert in the editorial office of the refereeing journalsZentralblatt für Mathematik and Fortschritte der Mathematik. At the same timehe was Privatdozent at Berlin University. From then on H. L. Schmid lived in thesame city as Grün and it is possible that they had met there occasionally. In anycase, H. L. Schmid knew about the mathematical background and the achievementsof Grün, and he knew what Grün needed: namely a quiet place to pursue his researchon group theory. This was what he could offer now, with remarkable consequencesfor Grün’s output of mathematical papers in the years to follow (see Section 1.5.4).

Grün’s salary at the Berlin Academy was not high, in fact it was quite small andjust enough to live on. But since Grün was single, this was acceptable to him.57

In October 1946 Grün had received an offer for a teaching position from theUniversity of Greifswald, as he narrates in his vita written August 2, 1955. However,they required there that he publicly committed himself to a political party in the Sowjetoccupation zone, and this he refused. Grün was a non-conformist: in the 1930s hehad refused to join the Nazi party, and now he did the same thing with the communistdominated parties.58

Perhaps we are not wrong to assume that there was another reason for Grün,conscious or unconscious, to reject this offer to Greifswald. For, he did not liketo teach. In fact, by all indications we know he was not a good lecturer. Andso he preferred to live on the small but sufficient income he got from the BerlinAcademy, free to pursue his studies on group theory without worrying about teachingand administrative or political problems.

54Cited from [JL98].55In a letter to Hasse dated July 1, 1946 Grün wrote: “I am relatively well off considering the circumstances.

I am working at the university but as a researcher only, which after all is what I wish to do.”. – After the war in1945, the “Friedrichs-Wilhelm Universität” of Berlin was short named “Universität Berlin”, and later in 1949 itwas renamed “Humboldt Universität zu Berlin”. It was situated in the Eastern (Soviet) sector of Berlin and is tobe distinguished from the “Free University” which had been founded in the Western sector.

56The documents of Grün’s employment at the Berlin Academy are preserved and available in the Academy’sarchive.

57H. L. Schmid was able to support also a number of other young (and not so young) mathematicians whoneeded help. One of them was Kurt Heegner, the man who later would be the first to solve the class number 1problem for imaginary quadratic fields [Hee52]. (Heegner’s paper was formulated in too fragmentary style andhence it was not understood properly until Deuring [Deu68] cleared up the situation.)

58Quite generally, people who knew him tell me that Grün’s opinions and beliefs were remarkably independentof the Zeitgeist.


Already in 1942 Hasse had written to Grün explaining what possibilities there werefor him to obtain his doctorate. But at that time nothing came out of this. Now in 1946H. L. Schmid proposed to Grün to apply to the university for admission to promotionfor doctorate. It is reported (by hearsay) that Grün was quite hesitating because hedid not like formalities of any kind. For, there had to be an extra permission becauseGrün had not been a student of Berlin University, in fact he had never attended anyuniversity. But H. L. Schmid finally succeeded to persuade Grün.59

Thus on April 2, 1946 Grün submitted the necessary application form to thedean of the science faculty of Berlin University. The fields in which he asked to beexamined were “Pure Mathematics, Applied Mathematics and Theoretical Physics”.He submitted the thesis Beiträge zur Gruppentheorie III (Contributions to grouptheory III) which two years later was published in the first volume of the new journalMathematische Nachrichten (See [Grü48a]). Officially H. L. Schmid signed as thefirst referee for the thesis but he mentioned in his report that Magnus, as an expert inthis field, had checked it thoroughly.

The promotion documents for Otto Grün are preserved at the archives of theHumboldt University. The examination took place on June 20, 1947 and the finaldoctor’s diploma is signed on September 20, 1948. At this date Grün was 60 years.

2.7.2 16 more papers

In Section 2.5.2 we have cited a letter of Grün (dated May 9, 1935) in which heclaimed to have “a huge pile of notes which already contain the essential ingredientsof future publications”. Some of those publications, until 1945, we have alreadymentioned. But it seems there was more in Grün’s pile of notes. For, from 1948 to1964 Grün published 16 more papers, 13 of them on p-groups and related topics.(The first of those papers he had used as his doctoral thesis.) About every year hecompleted a new paper. This activity seems quite remarkable, considering that Grünin 1948 was of age 60, and he was 76 at the time when his last paper appeared.

The first few of these papers were still checked by Magnus before publication,but later, Magnus had emigrated to USA, Grün was at last able to work on his own.He had learned to avoid erroneous conclusions in his publications and had become arespected colleague among group theorists. He wisely stayed away from great andfamous problems, in view of his experiences he had gone through in earlier years withVandiver’s conjecture, Burnside’s problem and the conjecture of Schur.60 His papersconstituted valuable and useful contributions for the specialists; they appeared ingood journals in Germany and elsewhere. Grün became a known expert in p-groupsand related structures, and he was consulted as a referee for doctorate theses etc.

59It is not unlikely that H. L. Schmid used the argument that if Grün had the title of “doctor” then this wouldimply some increase of his (small) salary.

60In 1938 Grün had published a paper [Grü38] in which he claimed (among other results) that every represen-tation of a finite group of exponent m can be realized in the field of m-th roots of unity. Schur had conjecturedthis in 1912 with the group order instead of exponent. However, Grün’s proof turned out to be erroneous.


Two of Grün’s papers from this time were on number theory: perfect numbers,and Bernoulli numbers. But these were only small notes.

In 1958 there was an increase in exchange of letters between Hasse and Grün,and this concerned class groups of cyclotomic fields. Thus Grün had not completelyforgotten this topic with which he had started in the 1930s. As a result of thiscorrespondence Grün obtained a theorem which, however, turned out to be a specialcase of Leopoldt’s Spiegelungssatz [Leo58]. Leopoldt’s paper was in press but notyet published. Hasse offered to publish Grün’s manuscript since, after all, it had beenobtained independently, but Grün withdrew his manuscript. Nonetheless his lettersshow that Grün’s number theoretical interest was still alive, and his standard washigh.

2.7.3 Würzburg (1954–1963)

The hope that Berlin would be able to establish itself as a center of Mathematicsin Germany dwindled soon. Around 1950 the “Gleichschaltung”, in the communistsense, of academic (and other) institutions in the Soviet occupied part of Germanywas intensified. As a consequence many people tried to go to West Germany. Hasseaccepted a position in Hamburg in 1950, and a number of younger people of hiscircle went with him. In 1953 H. L. Schmid changed from Berlin to the University ofWürzburg and again, a number of people went with him there.

Otto Grün too was among those who followed H. L. Schmid to Würzburg. Thelatter had been able to find means there for the financial support of Grün. At firstGrün became a member of the research center for applied mathematics in Würzburgwhich H. L. Schmid had newly founded together with Bilharz.61

Later, after the early death of H. L. Schmid in 1956, Grün could be supportedthrough a teaching job (“Lehrauftrag”) for group theory at the University ofWürzburg,which he received almost regularly for several years. There are still people livingwho have attended Grün’s lecture courses, or at least have tried to do so. The storyis that each semester Grün announced a lecture on group theory, and after 2–3 hoursevery student had dropped out because of Grün’s “awkwardness in the presentationof material” (which Hasse had already observed in 1939). After that, Grün was happyto be able to turn to his research without having to worry about lectures.

Between 1954 and 1961 Grün attended every group theory meeting in Oberwol-fach; these meetings were directed by Reinhold Baer, one of them by Jean Dieudonné.Since participation in Oberwolfach meetings is possible by personal invitation only,this shows that his results were appreciated by the international group theory com-munity. Four times Grün presented talks at those meetings (1955, 1959, 1960, 1961).

61Herbert Bilharz had been, like H. L. Schmid, a graduate student of Hasse. In his Göttingen thesis [Bil37] hehad solved Artin’s conjecture for primitive roots in the function field case – assuming the Riemann hypothesisfor function fields (which was finally verified by A. Weil). Later he went to applied mathematics and worked fora time in the aircraft industry. In Würzburg he held a chair for applied mathematics.


The abstracts of those talks are still available in the Oberwolfach abstract books (“Vor-tragsbücher”), they show that Grün talked about the results which he had obtainedin his papers. But as some participants of those meetings remember, his style oflecturing had not improved.

2.8 Epilogue

In 1955 Grün was 67 years. It became clear that something had to be done to secure forhim some retirement pension.62 This was difficult since he never had held a regularposition in a university. In the archives of Würzburg University I have found a numberof documents, between 1955 and 1962, written by the Mathematics Department Head,with the intention to obtain some kind of retirement pay for Grün.

In order to back those efforts, some leading group theorists were asked to writetheir opinion on Grün. Let us cite excerpts of those opinions, all dated in 1955, inorder to put into evidence that Grün was respected as a group theorist throughout theworld:

F.W. Levi, Freie Universität Berlin: Es ist Herrn Grün gelungen, neue Methodenfür die Erforschung der endlichen Gruppen zu entwickeln und dadurch diesesGebiet neu zu erschliessen. Schon seine ersten Ergebnisse haben Aufsehenunter den Algebraikern erregt und sind schnell in die Literatur, sogar in Lehr-bücher übergegangen. Seit dieser Zeit hat er unermüdlich weiter gearbeitet,wichtige Ergebnisse erzielt und dadurch anderen Mitarbeitern den Weg zuneuer Forschung geebnet. … Herr Grün ist Autodidakt, hat nie ein Lehramtbekleidet, aber er ist ein echter Gelehrter, und zwar ein Gelehrter von großerwissenschaftlicher Bedeutung.

Grün succeeded to develop new methods for the investigation of finite groupsand thus to open this field from a new viewpoint. Already his first results haveattracted great attention among algebraists and were quickly included into theliterature, even into textbooks. Since then he has ever continued to work, hehas obtained important results and thus opened the way for the research of othermathematicians. … Grün is self-educated, has never had a teaching position,but he is a true scholar with great scientific standing …

R. Baer, University of Illinois, Urbana: O. Grün ist unzweifelhaft einer der führen-den Gruppentheoretiker unserer Zeit. … In der fundamentalen Arbeit über dieendlichen p-Gruppen ist es ihm gelungen, die Ph. Hallsche Theorie der reg-ulären p-Gruppen auf beliebige p-Gruppen auszudehnen, den dabei entste-henden neuen Phänomenen Rechnung zu tragen und dadurch neues Licht aufdie Fülle der Erscheinungen in diesem reichen Gebiet zu werfen.

62In a letter of Grün to Hasse of August 29, 1955, Grün writes that he gets only 160 DM monthly.


Without doubt Grün is one of the leading group theorists of our time. … In thefundamental paper on finite p-groups he succeeded to extend Ph. Hall’s theoryof regular p-groups to p-groups of arbitrary structure. He was able to dealwith the new phenomena which showed up in this process, and thus to thrownew light upon the many aspects of this rich mathematical discipline.

B. H. Neumann, Hull: Otto Grün muss heutzutage als einer der bekanntesten undberühmtesten Gruppentheoretiker gelten, und zwar keineswegs nur in Deutsch-land, sondern überall, wo Mathematik getrieben wird … In drei so verschieden-artigen Monographien wie „Lehrbuch der Gruppentheorie“ von Zassenhaus,„Gruppi astratti“ von Scorza und „Teoriya Grupp“ von Kurosch werden dieResultate von Grün mehrfach herangezogen.

Nowadays Otto Grün has to be counted as one of the most prominent grouptheorists, by no means in Germany only but wherever mathematics is present… In three quite different monographys like “Lehrbuch der Gruppentheorie”by Zassenhaus, “Gruppi astratti” by Scorza and “Teoriya Grupp” by Kuroshhis results are repeatedly used.

J. Dieudonné, Evanston, Ill.: … confirmer tout l’estime et l’admiration que j’aipour les travaux de M. le Prof. O. Grün. Ses idées sur la théorie des groupes sedistinguent par une remarquable originalité et une profondeur peu commune …

… the estimation and admiration which I harbor for the works of Prof. O. Grün.His ideas about group theory are distinguished by a remarkable originality anda rarely found depth …

W. Magnus, New York University: Grün ist ein Mathematiker von wohlbegrün-detem internationalen Ansehen. Seine Arbeiten zur Gruppentheorie werdenvon mathematischen Autoren aller Länder zitiert, und einige der von HerrnGrün gefundenen Resultate gehören zum bleibenden Bestand der Gruppenthe-orie, was darin zum Ausdruck kommt, dass sie in allen modernen Lehrbücherndargestellt werden (z. Bsp. Zassenhaus, Kurosch) …

Grün is a mathematician of well founded international standing. His papers arecited by mathematical authors of all countries, and some of his results belongto the perpetual stock of group theory, which is evidenced by the fact that theyare treated in all modern textbooks (e.g., Zassenhaus, Kurosh) …

H. Zassenhaus, McGill University, Montreal: Im Bereiche der mathematischenForschung dieses Jahrunderts ist mir kein anderes Beispiel der Entdeckungeines hervorragenden Mathematikers im vorgerückten Alter bekannt gewor-den. Im neunzehnten Jahrhundert hat es die Fälle von Sophus Lie und Weier-strass gegeben … Durch seine Arbeiten hat sich Otto Grün einen Namen alsausgezeichneter tiefforschender deutscher Mathematiker gemacht, den ich in


England und in den Vereinigten Staaten immer wieder mit Achtung und Be-wunderung habe nennen hören.

In the realm of mathematical research I do not know any other example of anexcellent mathematician who was discovered in his midlife years only. In the19th century there were the cases of Sophus Lie and Weierstrass … Through hiswork Otto Grün has become a well known name as a German mathematician,doing deep research. I have heard mention his name again and again in Englandand in the United States with respect and admiration …

It is not clear from the Würzburg documents whether the initiative on behalf ofGrün was successful. I am afraid it was not.

In any case, Grün returned to (West-)Berlin, his home town, in the year 1963 whenhe was 75. After that date there were still some letters exchanged between Grün andHasse but they were restricted mainly to birthday greetings and the like. All the timeGrün continued to respect Hasse as his teacher, the one who opened mathematics forhim, and he expressed his thanks and admiration for Hasse in his letters.

Starting from 1971 we find in Grün’s letterhead the title of “Professor”. Perhapswe can conclude from this that he had obtained from the government this official titleand, we hope, finally some adequate retirement pension in view of his achievements.

In October 1974 Grün died at the age of 86. Among Hasse’s papers I found abrief obituary, about half a page, dated October 10, 1974. But I do not know whereit had been published; perhaps it was a newspaper clip. There was no obituary in theJahresbericht of the DMV of which Grün was a member since 1939.

2.9 Addendum

I am indebted to Prof. Siegmund-Schultze who, after the above article had appeared,had sent me a message dated January 9, 2006 with the information that the Bundes-archiv in Berlin contains a file on Otto Grün.63 In the meantime I have been able tolook at it.

That file starts with a letter from Grün dated November 4, 1936 to the REM statingthat he was without any income and asking for help to find a position. The letter isadressed to the mathematician Theodor Vahlen who, at that time, was a powerfulfigure in the REM. In reply to Grün’s letter Vahlen asked him for a curriculum vitaeand a list of publications, and Grün supplied this eventually.

From his curriculum vitae we learn some facts about Grün’s former commercialactivities: Until 1925 he had been the manager of the Berlin branch of an Austrianwinery, thereafter he had been working as an independent auditor.

63REM 2634. The abbreviation “REM” stands for Reichserziehungsministerium, i.e., the Ministerium foreducational policy of the German government which had been established by the Nazi government. In thefollowing we shall use this abbreviation for short.


The REM then asked Hasse, Bieberbach and Tornier for their opinion on Grün.Hasse, as we have seen, knew Grün from earlier correspondence and from Grün’svisit to Göttingen in 1935. He sent an extended opinion saying that Grün is talented,has already obtained siginifant results, and seems to promise further success in hisresearch activities. But he adds that Grün, having no formal academic training, hasproblems in teaching; he is not able to present his material in a clear way. Also, hismathematical knowledge is very narrow. Therefore, Hasse continues, he recommendsto give Grün a grant for continuing his reasearch, preferably in connection with auniversity. Then one could see whether he would develop to become a useful memberof the teaching staff.

Bieberbach wrote that he had not known Grün but has invited him for an interview.From this he got the impression that Grün had acquired unusual knowledge in algebraand number theory and was familiar with the traditional problems and methods. Butin Bieberbach’s opinion it would be too early to decide whether a research grantwould be appropriate. Moreover, he added, Grün’s mathematical interests were quiteone-sided and so he would be of no use for, say, working on mathematical problemsof aircraft construction. He proposed to employ Grün in some administrative positionin such a way that there would be sufficient time for him to follow his mathematicalinterests. Bieberbach also mentioned that his letter was written in accordance withTornier who had been present at the interview with Grün.64

The next document is an entry of the REM, signed by Dr. Dames on December 14,1936, reporting that he had met Grün and informed him that he will get some financialsupport for his research work, at least for the next months. (A somewhat modestamount of Reichsmark was mentioned.) Grün was advised for a possible job toget into contact with Professor Bartels, the director of the Geophysical Institute inPotsdam.

Apparently, for such a job Grün had to provide evidence for his “arian descent”which was required in Nazi Germany. It took some time for him to provide thatdocument, and so he could start his work in Potsdam in January 1938 only. Fromthen on, every year in the month of March there was an appplication, sent by Grün,for continuing his job at the Potsdam Institute, and this was supported by the directorof the Institute. According to these documents Grün did certain analytic and numericcomputations which arose in the work of the Institute, e.g., in order to describehomogeneous magnetic fields. The last of those documents is dated March 1944, sothat apparently Grün’s financial situation was secured until March 1945. But we havefound no evidence that Grün had won enough freedom during that period to continuehis group theoretical research.

64This corroborates the fact that E. Tornier had been removed from his professorship in Göttingen in view ofhis personal conduct, and that he is now at Berlin University. But since he posed as a staunt Nazi he still seems toenjoy the protection of Bieberbach and of Vahlen. (According to the files at the archive of Humboldt-Universityin Berlin, this changed during the year 1937 when it became known that Tornier was addicted to alcohol (anddrugs) and was involved in financial fraudulence; he then was removed from the university.)

Chapter 3

At Emmy Noether’s funeral

Translation of the article:

Zu Emmy Noethers Geburtstag. Einige neue Noetheriana.

Mitteilungen der Deutschen Mathematiker Vereinigung 15/1 (2007), 15–21.

3.1 Introduction 1173.2 Funeral speech by Hermann Weyl 1233.3 Grete Hermann to van der Waerden 1253.4 President Park to Otto Noether 1263.5 Marguerita Lehr, Professor at Bryn Mawr College 127

3.1 Introduction

The day of March 23, 2007 marks the 125th anniversary of Emmy Noether’s birthday.It may be suitable on this occasion to present in this article some new “Noetheriana”,commemorating this great master of our science. I shall present some documentswhich may help us to understand her life and the impact of her ideas on mathematicsand mathematicians.

First, I will show some documents from the Nachlass of Grete Hermann:

1. The text of a short speech of Hermann Weyl at Emmy Noether’s funeral, de-livered on April 17, 1935 in Bryn Mawr. The speech, which we present herein English translation, was delivered in German language and was addressedto a small circle of mourners.

2. A letter of Grete Hermann (in English translation) addressed to van der Waer-den, dated January 24, 1982, in which she remembers their common days asstudents of Emmy Noether. In this letter she mentions also the text of the abovementioned speech by Hermann Weyl.

Secondly, I will show some documents from the archive of Bryn Mawr College whereEmmy Noether had found shelter after her forced emigration from Germany:

118 3 At Emmy Noether’s funeral

3. A report by Marion Park, Ph.D., at that time the president of Bryn MawrCollege, about the funeral ceremony mentioned above. The report is addressedto Emmy Noether’s cousin Otto Noether in Mannheim.

4. The text of an address of Professor Marguerita Lehr from Bryn Mawr, readone day after the above mentioned funeral, on April 18, 1935, in the Chapelof Bryn Mawr. Lehr reports about the reception of Emmy Noether by facultyand students of Bryn Mawr, and her activities there.

I do not claim that these four documents are of special importance. But theygive us at least some evidence about the impact which Emmy Noether had left onher environment. For we still have the following question unanswered: How can weexplain the fact that Emmy Noether had been able to exert, during her lifetime, sucha wide influence on the mathematical thinking of her contemporaries?

After all, she had comparatively few publications: her Collected Papers amountto just one volume, and only less than half of it contains those papers which areusually cited as witnesses of her fame. And even those were often written not byherself but by her students (van der Waerden, Deuring). Also, she was not a brilliantlecturer; all accounts of her contemporaries tell us that her lectures have to be rated aschaotic, according the usual criteria. The lectures were comprehensible to the smallselected circle of disciples only, who were used to her style of talking. Her wayto do mathematics was not appreciated in every corner. For instance, Olga Tausskyreports that even some of Noether’s Göttingen colleagues criticized the abstract formin which she expressed her ideas. And this still happened in the early 1930s, i.e., ata time when her name was already known worldwide and the attraction of Göttingenas a mathematical center rested mainly upon her fame.1 But then, what made her riseto a unique personality among the mathematicians of the 20th century?

On first sight the answer to this question may be that she represented a math-ematical trend which was to spread anyhow, namely “Modern Algebra” and, moregenerally, the “modern abstract way of reasoning”. But this seems to be short ofproviding a meaningful explanation of the whole phenomenon. Certainly her uniquepersonal character was involved.

Unfortunately we have only few direct records about her. There is no film, thereare only a few photos and no interviews. There do exist meaningful reports writtenby contemporaries, but not many. Hence every new document about her, and abouther environment, may let us catch a glimpse of some characteristic trait of her, andso may serve as a contribution in forming a valid picture.

Before presenting those documents it is perhaps not without interest to reportabout how I discovered them; this happened not without some luck.

All of this started with our plan to edit the correspondence between Hasse andEmmy Noether [LR06]. Those letters document in rare clarity the development of the

1Compare the testimonials cited in Chapter 5.


mathematical relation between the two. Hasse was not one of the so-called “Noetherboys” who had gathered around her in Göttingen. He had been raised in a completelydifferent mathematical environment, viz. in Marburg with Hensel. As we learn fromthe letters it happened in 1924, when Hasse had the position as Privatdozent at theUniversity in Kiel, that he came into the circle of influence of Emmy Noether. He mether at the annual meeting of the DMV (German Mathematical Society) in Innsbruckwhere she presented her axiomatic description of what today are called Dedekindrings. Hasse was impressed about the easiness and the generality of Noether’s theory.From the letters we learn that he became more and more convinced of the greatpower which was inherent in Emmy Noether’s conception of mathematics2 (althoughthe contact with Emmy Noether never led Hasse to give up his own mathematicalindividual style).

The influence of Emmy Noether, on Hasse as well as on others, was based on herability to formulate their problems in an abstract form which, in her opinion, clarifiedthe situation. She did not solve mathematical problems but she led the way to thesolution by putting them on an abstract track which, in her opinion, would lead tothe solution by simplification.3 We observe this not only in her relation with Hassebut also, e.g., with van der Waerden, with Alexandrov and to a certain degree withHermann Weyl.

Hermann Weyl, three years younger than Emmy Noether, reports:

I have a vivid recollection of her when I was in Göttingen as visitingprofessor in the winter semester of 1926–1927, and lectured on repre-sentations of continuous groups. She was in the audience; for just atthat time the hypercomplex number systems and their representationshad caught her interest and I remember many discussions when I walkedhome after the lectures, with her and von Neumann … through the cold,dirty, rain-wet streets of Göttingen. When I was called permanently toGöttingen in 1930, I earnestly tried to obtain from the Ministerium a bet-ter position for her, because I was ashamed to occupy such a preferredposition beside her whom I knew to be my superior as a mathematician inmany respects … In my Göttingen years, 1930–1933, she was withoutdoubt the strongest center of mathematical activity there, consideringboth the fertility of her scientific research program and her influenceupon a large circle of pupils.4

How can it be explained that one of the leading mathematician of the time lookedat Emmy Noether as his “superior” ? After all, Emmy Noether did not have a positionas professor in Göttingen but only as a lecturer with very small remuneration, and

2See, e.g., his invited lecture 1929 in Prague with the title “Die algebraische Methode”. [Has30b].3See, e.g., the obituary, written by van der Waerden [vdW35].4Cited from the published Memorial Address of Weyl for Emmy Noether, which he delivered on April 26,

1935 at Goodhart Hall in Bryn Mawr. This address is printed in the Noether biography of Auguste Dick [Dic70].


this position was granted only for one year at a time and had to be applied foreach year again. There were many other brilliant professors and lecturers at themathematics department in Göttingen, but Weyl did not say anyting like this aboutother people. From the letters between Emmy Noether and Hasse we can concludethat an important ingredient in their relation was Emmy’s ability, not only to distributenew and inspiring ideas for future development, but also to establish warm personalcontacts to her colleagues and pupils. And this applies of course not only to Hassebut also to others, in particular to Hermann Weyl. This personal note, paired withher insistent persuasive power, can be observed in all reports which we have aboutEmmy Noether.

Emmy Noether died on April 14, 1935. There exists a letter of Hermann Weyl toHasse, dated April 30, 1935, in which he reports about the small funeral ceremonywhich had been held on April 17, 1935. Weyl wrote that the wreath of the Göttingenmathematicians had been placed visibly on the coffin, as Hasse had wished it to be.Moreover, Weyl wrote:

The friends in Germany can be assured that everything was done hereto offer the deceased a dignified farewell.

Weyl added that he includes in the letter a copy of his short speech which hedelivered at the funeral. But at the end of the letter he wrote:

P.S. After all, it appears to me more prudent not to include the text ofmy speech. H.W.

Weyl does not explain why he did not include that text but looking at the date we canguess why – it was the year 1935, two years after the Nazis had come to power.

We have searched for the text of Weyl’s speech, since we wished to present it inour book of the Hasse–Noether correspondence. But we did not find it in the papersleft by Weyl, nor in those of Richard Brauer who was close to Emmy Noether in hisyears in Princeton, and also not in the papers of other friends of hers as, e.g., vander Waerden. The text of Weyl’s longer address which he delivered a week later atthe official memorial in Bryn Mawr has been published several times and hence isavailable to the mathematical community. Therefore we did not reproduce it againin our book. We only informed the reader in a footnote that the text of Weyl’s shortspeech at the funeral seemed to be lost.

However, recently I found that text quite unexpectedly at a place where we hadnot expected it to be. But it was not possible any more to include it in our book. Thatis the reason why I am presenting it here to the mathematical public.

It was quite by accident that I was led to that text. While reading the autobiographyof the physicist Werner Heisenberg “Der Teil und das Ganze” [Hei69]. I found achapter where he reported on a seminar in Leipzig in which he discussed with other


colleagues the philsophical foundations of the new quantum theory, which had beenestablished by his own essential contributions. There he says:

A special opportunity for philosophical discussion came one or twoyears later when a young lady philosopher, Grete Hermann, joined us inLeipzig in order to dispute with the nuclear physicists their philosophicalstatements.

In the following chapter Heisenberg reports on the content of those discussions withGrete Hermann (and with Carl Friedrich von Weizsäcker), which took place in theyear 1934.

Upon reading the name of Grete Hermann my curiosity was aroused. Was thispossibly the same Grete Hermann who had been in 1925 the first doctoral studentof Emmy Noether, with a thesis about polynomial operations in finitely many steps?In this connection her name used to be well known among people doing numbertheory; her paper in the Mathematische Annalen is still of interest (including van derWaerden’s comments in the next volume of Mathematische Annalen). However, Idid not know anything about her life. (After all, it happens quite often that peopleknow the name and perhaps the most important results of mathematicians of formergenerations, but much less about their personal life.)

Heisenberg had reported that the said seminar discussions took place in the townof Leipzig. This reminded me that, at that time, van der Waerden had been profes-sor at Leipzig University. We know from his own words that he had accepted thatposition particularly since there he could establish contact with Heisenberg and thepeople around him. Van der Waerden’s book Die gruppentheoretische Methode in derQuantenmechanik (Group theoretical methods in quantum mechanics) documents hisconnection to the theoretical physicists in Leipzig of that time. In Göttingen van derWaerden had been a fellow student of Grete Hermann. Was it possible that she hadcome to Leipzig through her contact with van der Waerden?

My inquiries then showed definitely that the Grete Hermann who was mentionedin Heisenberg’s book was identical with the Grete Hermann who had been a student ofEmmy Noether. Here we cannot go into the interesting and quite unusual biographyof Grete Hermann, although this would be a worthwhile task, of interest not only formathematicians but also for physicists, philosophers, social scientists and educators.But I learned that there exists a Nachlass of Grete Hermann in the “Archiv der sozialenDemokratie” in Bonn.

I went to that archive hoping to find letters from Emmy Noether but was disap-pointed. In retrospect this is understandable since while in Göttingen they did notcommunicate by letters but they talked with each other.

However, quite unexpectedly I found there the German text of Weyl’s speech.It seems that although Weyl did not send the text to Hasse he had sent it to GreteHermann. Or, maybe she obtained this text at some later time, perhaps through vander Waerden? We do not know.


In any case, I believe that this moving text should not be forgotten. We feel thatthere was more than high esteem for Emmy Noether as a colleague. She had alsobeen able to establish a special personal relationship to Weyl. He was deeply movedby her sudden, unexpected death, and this enabled him to present such a speech in aconvincing manner.

We also show a letter of Grete Hermann to van der Waerden which we found inher Nachlass too. As we see, the speech of Weyl is mentioned there. In that Nachlassthere are also two letters of Auguste Dick (who had written the first and only Noetherbiography) to Grete Hermann, but we did not include it in our article.

A first version of our article had been submitted already in September 2007 butonly with the first two documents cited at the start of this chapter. Since then I havediscovered in various archives more “Noetheriana” which in my opinion also wouldbe of interest. Those are planned to be published later, except the two which I havefound in the archive of Bryn Mawr and which refer explicitly to the funeral of EmmyNoether.

The report of President Park, mentioned in 3. above, is part of a letter whichshe wrote to Dipl. Ing. Otto Nöther, a cousin of Emmy Noether, who resided in theGerman town of Mannheim. The letter contains, among others, a precise medicalreport about the cause of the death of Emmy Noether following a tumor operation.It also contains details about the Nachlass of Emmy Noether. Here we have onlyincluded that part of this letter which concerns her funeral. We read that not onlyHermann Weyl had delivered a short speech but also Richard Brauer and Olga Tausskyand, in addition, Anna Pell Wheeler, Head of the Mathematics Department in BrynMawr.

At that time Richard Brauer held a position as assistant to Hermann Weyl at theInstitute for Advanced Study in Princeton. During Emmy Noether’s frequent visits toPrinceton there had developed a close relationship between her and the Brauer family.

Olga Taussky had been introduced to Emmy Noether during her time in Göttingenas a post-doc. In 1935 she worked at Bryn Mawr college as a lecturer.

Anna Wheeler had studied in Göttingen with Hilbert in the years 1906–1908.There she had written her thesis on integral equations but she did not get her Ph.D.in Göttingen. (“I had some trouble with Professor Hilbert”.) It was only in 1910that she got her doctoral degree in Chicago with E. H. Moore. We know from varioussources that in Bryn Mawr a friendly relationship developed between Anna Wheelerand Emmy Noether.

The text of the speeches of Brauer, Taussky and Wheeler seem not to have beenpreserved. On the other hand, I found in the archives of Bryn Mawr a handwrittenmanuscript by Marguerita Lehr, professor in the mathematics department, in whichshe reports on the reception of Emmy Noether in Bryn Mawr. The text is dated:“Chapel – Thursday April 18, 1935”, i.e., one day after the funeral. We suppose thatit was read at a service in the chapel of Bryn Mawr on that day. The text describes,from the point of view of her colleagues in Bryn Mawr, how Emmy Noether had been


received and how she had started her new academic life there. In a short time EmmyNoether had won the hearts of her colleagues and students. I believe this is of interestin addition to the reports which she had sent to Hasse from Bryn Mawr.

At the end of her manuscript Lehr writes that Emmy Noether’s “two years at BrynMawr were happy ones” and for this she refers to statements of friends. In connectionwith this we have found a letter of Abraham Flexner, the founder and first director ofthe Institute for Advanced Study in Princeton; the letter is dated April 25, 1934 andis addressed to President Park. There we read:

… it ought to make you and Mrs. Wheeler happy to know that a fewweeks ago she [Emmy Noether] remarked to Professor Veblen that thelast year and a half had been the very happiest in her whole life, forshe was appreciated in Bryn Mawr and Princeton as she had never beenappreciated in her own country.

Now, here are the announced documents:5

3.2 Funeral speech by Hermann Weyl

The following text6 was read out loud by Hermann Weyl on Emmy Noether’s funeralon April 17, 1935.

The hour has come, Emmy Noether, in which we must forever take our leave ofyou. Many will be deeply moved by your passing, none more so than your belovedbrother Fritz, who, separated from you by half the globe, was unable to be here, andwho must speak his last farewell to you through my mouth. His are the flowers I layon your coffin. We bow our heads in acknowledgement of his pain, which it is notours to put into words.

But I consider it a duty at this hour to articulate the feelings of your Germancolleagues – those who are here, and those in your homeland who have held true toour goals and to you as a person. I find it apt, too, that our native tongue be heard atyour graveside – the language of your innermost sentiments and in which you thoughtyour thoughts – and which we hold dear whatever power may reign on German soil.Your final rest will be in foreign soil, in the soil of this great hospitable country thatoffered you a place to carry on your work after your own country closed its doors onyou. We feel the urge at this time to thank America for what it has done in the lasttwo years of hardship for German science, and to thank especially Bryn Mawr, wherethey were both happy and proud to include you amongst their teachers.

5With friendly permission of the “Archiv der sozialen Demokratie” in Bonn and the archive of Bryn MawrCollege.

6Translated from German by Ian Beaumont.


Justifiably proud, for you were a great woman mathematician – I have no reser-vations in calling you the greatest that history has known. Your work has changedthe way we look at algebra, and with your many gothic letters you have left yourname written indelibly across its pages. No-one, perhaps, contributed as much asyou towards remoulding the axiomatic approach into a powerful research instrument,instead of a mere aid in the logical elucidation of the foundations of mathematics, asit had previously been. Amongst your predecessors in algebra and number theory itwas probably Dedekind who came closest.

When, at this hour, I think of what made you what you were, two things immedi-ately come to mind . The first is the original, productive force of your mathematicalthinking. Like a too ripe fruit, it seemed to burst through the shell of your humanness.You were at once instrument of and receptacle for the intellectual force that surgedforth from within you. You were not of clay, harmoniously shaped by God’s artistichand, but a piece of primordial human rock into which he breathed creative genius.

The force of your genius seemed to transcend the bounds of your sex – and inGöttingen we jokingly, but reverentially, spoke of you in the masculine, as “denNoether”. But you were a woman, maternal, and with a childlike warmheartedness.Not only did you give to your students intellectually – fully and without reserve –they gathered round you like chicks under the wings of a mother hen; you loved them,cared for them and lived with them in close community.

The second thing that springs to mind is that your heart knew no malice; youdid not believe in evil, indeed it never occurred to you that it could play a role inthe affairs of man. This was never brought home to me more clearly than in thelast summer we spent together in Göttingen, the stormy summer of 1933. In themidst of the terrible struggle, destruction and upheaval that was going on around usin all factions, in a sea of hate and violence, of fear and desperation and dejection– you went your own way, pondering the challenges of mathematics with the sameindustriousness as before. When you were not allowed to use the institute’s lecturehalls you gathered your students in your own home. Even those in their brown shirtswere welcome; never for a second did you doubt their integrity. Without regard foryour own fate, openhearted and without fear, always conciliatory, you went your ownway. Many of us believed that an enmity had been unleashed in which there couldbe no pardon; but you remained untouched by it all. You were happy to go back toGöttingen last summer, where, as if nothing had happened, you lived and workedwith German mathematicians striving for the same goals. You planned on doing thesame this summer.

You truly deserve the wreath that the mathematicians in Göttingen have asked meto lay on your grave.

We do not know what death is. But is it not comforting to think that souls willmeet again after this life on Earth, and how your father’s soul will greet you? Hasany father found in his daughter a worthier successor, great in her own right?


You were torn from us in your creative prime; your sudden departure, like the echoof a thunderclap, is still written on our faces. But your work and your dispositionwill long keep your memory alive, in science and amongst your students, friends andcolleagues.

Farewell then, Emmy Noether, great mathematician and great woman. Thoughdecay will take your mortal remains, we will always cherish the legacy you left us.

Hermann Weyl

Emmy Noether’s tomb in Bryn Mawr

3.3 Grete Hermann to van der Waerden

Grete Henry-Hermann 2800 Bremen 1, den 24. Januar 1982Am Barkhof 19

HerrnProf. Dr. B. L. van der WaerdenWiesliacher 58053 Zürich

Dear Herr van der Waerden!

Please permit me to address you in this simple manner; this reflects my thoughtsback to our common time as students! In front of me there is the invitation toa collquium in Erlangen named after Emmy Noether. There you will talk about


your “Göttingen years of study”. This aroused so many memories: In my mind Iam seeing the lecture room No. 16 in the second floor of the Göttingen auditorium;Emmy Noether stands at the blackboard pondering intensively about something; infront of her only a small group of students is sitting who are heavily involved; youand I are among them.

I shall not come to the colloquium – not only because my ears which have grownold do not any more do their full duty, but mainly because in the course of thepast decades I have lost the contact to ideal theory. Already Emmy Noether saidresentfully to me, when after my examinations I became assistant to my other teacherin Göttingen, the philosopher Leonard Nelson: “Now she has studied mathematicsfor four years and suddenly she discovers her philosophical heart!”

But in this year, which contains the 100. birthdays of the two most importantteachers of my student years, I remember joyfully and thankfully this woman, ofmathematical originality and human affection. She lent me her help not only inmathematics but also with some annoying formal problems connected with exami-nations.

On the blackboard in the lecture room no. 16 she developed the many ideals asmentioned in one of her obituaries: “With many gothic letters you have left yourname written indelibly across the pages of mathematics.”7

In remembrance of Emmy Noether I am greeting you!YoursG.H.

3.4 President Park to Otto Noether

Extract from a letter dated May 18, 1935.

We arranged to hold the funeral at my own house on Wednesday, the 17th. Theservice at my house took place at three o’clock in the afternoon. There were, I think,about sixty persons present, all the members of faculty of Bryn Mawr College whohad known her, and her students here, and a large number of faculty and students ofthe Department of Mathematics at Princeton University. The coffin was completelycovered with beautiful flowers sent by many friends and organizations … The ex-ercises were very simple, and I thought beautiful. A trio of violin, cello and pianomusic of Bach and Mozart was played for ten minutes at the beginning and again atthe end of the service. Professor Wheeler and Dr. Olga Taussky spoke briefly onbehalf of Bryn Mawr College in English. Professor Weyl and Dr. Brauer spoke onbehalf of her German colleagues and friends in German …

7Here Grete Hermann obviously refers to the text of Weyl’s speech.


3.5 Marguerita Lehr, Professor at Bryn Mawr College

Text of the speech, handwritten, dated April 18, 1935.

When Bryn Mawr opened in 1933, President Park announced the coming of amost distinguished foreign visitor to the Faculty, Dr. Emmy Noether. Among math-ematicians that name always brings a stir of recognition; the group in this vicinitywaited with excitement and many plans for Dr. Noether’s arrival. At Bryn Mawr therewas much discussion and rearrangement of schedule, so that graduate students mightbe free to read and consult with Miss Noether until she was ready to offer definitelyscheduled courses. For many reasons it seemed that a slow beginning might haveto be made; the graduate students were not trained in Miss Noether’s special field,– the language might prove a barrier –, after the academic upheaval in Göttingenthe matter of settling into a new and puzzling environment might have to be takeninto account. When she came, all of these barriers were suddenly non-existent, sweptaway by the amazing vitality of the woman whose fame as the inspiration of countlessyoung workers had reached America long before she did. In a few weeks the classof four graduates was finding that Miss Noether could and would use every minuteof time and all the depth of attention that they were willing to give. In this secondyear her work had become an integral part of the department; she had taken on anhonors student, her group of graduates has included three research fellows here onscholarships or fellowships specially awarded to take full advantage of her presence,and the first Ph. D. dissertation directed at Bryn Mawr by Miss Noether has just goneto the Committee bearing her recommendation.

Professor Brauer in speaking yesterday of Miss Noether’s powerful influenceprofessionally and personally among the young scholars who surrounded her in Göt-tingen said that they were called the Noether family, and that when she had to leaveGöttingen, she dreamed of building again somewhere what was destroyed then. Werealize now with pride and thankfulness that we saw the beginning of a new “Noetherfamily” here. To Miss Noether her work was as inevitable and natural as breathing,a background for living taken for granted; but that work was only the core of herrelation to students. She lived with them and for them in a perfectly un-selfishnessway. She looked at the world with direct friendliness and unfeigned interest, and shewanted them to do the same. She loved to walk, and many a Saturday with five or sixstudents she tramped the roads with a fine disregard for bad weather. Mathematicalmeetings at the University of Pennsylvania, at Princeton, at NewYork, began to watchfor the little group, slowly growing, which always brought something of the freshnessand buoyance of its leader.

Outside of the academic circle, Miss Noether continually delighted her Americanfriends by the avidity with which she gathered information about the American en-vironment. She was proud of the fact that she spoke English from the very first; shewanted to know how things were done in America, whether it were giving a tea or


taking a Ph. D., and she attacked each single subject with the disarming candor andvigorous attention which won every one who knew her.

Emmy Noether might have come to America as a bitter person, or a despondentperson. She came instead in open friendliness, pleased beyond measure to go onworking as she had, even in circumstances so different from the ones she had loved.And our final consolation is that she made here too a place that was hers alone. Wefeel not only greatly honoured that she wanted to stay and work with us; we feelprofoundly thankful for the assurance that her friends have brought to us – that hertwo short years at Bryn Mawr were happy ones.

Chapel – Thursday April 18, 1935 Marguerita Lehr

Chapter 4

Emmy Noether and Hermann Weyl

Emmy Noether and Hermann Weyl.

In: Groups and Analysis: The legacy of Hermann Weyl, ed. by Karin Tent.

London Mathematical Society Lecture Note Series Vol. 394 (2008), 285–326.

This is the somewhat extended manuscript of a talk presented at the Hermann Weyl conferencein Bielefeld, September 10, 2006.

4.1 Preface 1294.2 Introduction 1304.3 The first period: until 1915 1314.4 The second period: 1915–1920 1364.5 The third period: 1920–1932 1384.6 Göttingen exodus: 1933 1494.7 Bryn Mawr: 1933–1935 1534.8 The Weyl–Einstein letter to the NYT 1564.9 Appendix: documents 159

4.1 Preface

This is a conference in honor of Hermann Weyl and so I may be allowed, beforetouching the main topic of my talk, to speak about my personal reminiscences ofhim.

It was in the year 1952. I was 24 and had my first academic job at München whenI received an invitation from van der Waerden to give a colloquium talk at ZürichUniversity. In the audience of my talk I noted an elder gentleman, apparently quiteinterested in the topic. Afterwards – it turned out to be Hermann Weyl – he approachedme and proposed to meet him next day at a specific point in town. There he told methat he wished to know more about my doctoral thesis, which I had completed twoyears ago already but which had not yet appeared in print. Weyl invited me to join himon a tour on the hills around Zürich. On this tour, which turned out to last for severalhours, I had to explain to him the content of my thesis which contained a proof of the

130 4 Emmy Noether and Hermann Weyl

Riemann hypothesis for function fields over finite base fields. He was never satisfiedwith sketchy explanations, his questions were always to the point and he demandedevery detail. He seemed to be well informed about recent developments.

This task was not easy for me, without paper and pencil, nor blackboard and chalk.So I had a hard time. Moreover the pace set by Weyl was not slow and it was notquite easy to keep up with him, in walking as well as in talking.

Much later only I became aware of the fact that this tour was a kind of examination,Weyl wishing to find out more about that young man who was myself. It seems thatI did not too bad in this examination, for some time later he sent me an applicationform for a grant-in-aid from the Institute for Advanced Study in Princeton for theacademic year 1954/55. In those years Weyl was commuting between Zürich andPrinceton on a half-year basis. In Princeton he had found, he wrote to me, that therewas a group of people who were working in a similar direction.

Hence I owe to Hermann Weyl the opportunity to study in Princeton. The twoacademic years which I could work and learn there turned out to be important formy later mathematical life. Let me express, posthumously, my deep gratitude andappreciation for his help and concern in this matter.

The above story shows that Weyl, up to his last years, continued to be activehelping young people find their way into mathematics. He really cared. I did notmeet him again in Princeton; he died in 1955.

Let us now turn to the main topic of this talk as announced in the title.

4.2 Introduction

Both Hermann Weyl and Emmy Noether belonged to the leading group of mathe-maticians in the first half of 20th century, who shaped the image of mathematics aswe see it today.

Emmy Noether was born in 1882 in the university town of Erlangen, as the daugh-ter of the renowned mathematician Max Noether. We refer to the literature for infor-mation on her life and work, foremost to the empathetic biography by Auguste Dick[Dic70] which has appeared in 1970, the 35th year after Noether’s tragic death. Itwas translated into English in 1981. For more detailed information see, e.g., the verycarefully documented report by Cordula Tollmien [Tol90]. See also Kimberling’spublications on Emmy Noether, e.g., his article in [BS81].

When the Nazis had come to power in Germany in 1933, Emmy Noether was dis-missed from the University of Göttingen and she emigrated to the United States. Shewas invited by Bryn Mawr College as a visiting professor where, however, she stayedand worked for 18 months only, when she died on April 14, 1935 from complicationsfollowing a tumor operation.1

1See footnote 48.


Quite recently we have found the text, hitherto unknown, of the speech whichHermann Weyl delivered at the funeral ceremony for Emmy Noether on April 17,1935.2 That moving text puts into evidence that there had evolved a close emotionalfriendship between the two. There was more than a feeling of togetherness betweenimmigrants in a new and somewhat unfamiliar environment. And there was more thanhigh esteem for this women colleague who, as Weyl has expressed it3, was “superiorto him in many respects”. This motivated us to try to find out more about their mutualrelation, as it had developed through the years.

We would like to state here already that we have not found many documents forthis. We have not found letters which they may have exchanged.4 Neither did EmmyNoether cite Hermann Weyl in her papers nor vice versa5. After all, their mathe-matical activities were going into somewhat different directions. Emmy Noether’screative power was directed quite generally towards the clarification of mathematicalstructures and concepts through abstraction, which means leaving all unnecessaryentities and properties aside and concentrating on the essentials. Her basic work inthis direction can be subsumed under algebra, but her methods eventually penetratedall mathematical fields, including number theory and topology.

On the other side, Hermann Weyl’s mathematical horizon was wide-spread, fromcomplex and real analysis to algebra and number theory, mathematical physics andlogic, also continuous groups, integral equations and much more. He was a mathe-matical generalist in a broad sense, touching also philosophy of science. His mathe-matical writings have a definite flair of art and poetry, with his book on symmetry asa culmination point [Wey52].

We see that the mathematical style as well as the extent of Weyl’s research workwas quite different from that of Noether. And from all we know the same can be saidabout their way of living. So, how did it come about that there developed a closerfriendly relationship between them? Although we cannot offer a clear cut answer tothis question, I hope that the reader may find something of interest in the followinglines.

4.3 The first period: until 1915

In the mathematical life of Emmy Noether we can distinguish four periods.6 In herfirst period she was residing in Erlangen, getting her mathematical education andworking her way into abstract algebra guided by Ernst Fischer, and only occasionallyvisiting Göttingen. The second period starts in the summer of 1915 when she came

2See [Roq07b]. We have included in the appendix an English translation of Weyl’s text; see Section 4.9.2.3See [Wey35].4With one exception; see Section 4.5.3.5There are exceptions; see Section 4.4.6Weyl [Wey35] distinguishes three epochs but they represent different time intervals than our periods.


to Göttingen for good, in order to work with Klein and Hilbert. This period iscounted until about 1920. Thereafter there begins her third period, when her famouspaper Idealtheorie in Ringbereichen (Ideal theory in rings) appeared, with which she“embarked on her own completely original mathematical path” – to cite a passagefrom Alexandrov’s memorial address [Ale83]. The fourth period starts from 1933when she was forced to emigrate and went to Bryn Mawr.

4.3.1 Their mathematical background

Hermann Weyl, born in 1885, was about three years younger than Emmy Noether.In 1905, when he was 19, he entered Göttingen University (after one semester inMünchen). On May 8, 1908 he obtained his doctorate with a thesis on integralequations, supervised by Hilbert.

At about the same time (more precisely: on December 13, 1907) Emmy Noetherobtained her doctorate from the University of Erlangen, with a thesis on invariantssupervised by Gordan. Since she was older than Weyl we see that her way to Ph.D.was longer than his. This reflects the fact that higher education, at that time, wasnot as open to females as it is today; if a girl wished to study at university and get aPh.D. then she had to overcome quite a number of difficulties arising from tradition,prejudice and bureaucracy. Noether’s situation is well described in Tollmien’s article[Tol90].7

But there was another difference between the status of Emmy Noether and Her-mann Weyl at the time of their getting the doctorate.

On the one side, Weyl was living and working in the unique Göttingen mathe-matical environment of those years. Weyl’s thesis belongs to the theory of integralequations, the topic which stood in the center of Hilbert’s work at the time, andwhich would become one of the sources of the notion of “Hilbert space”. And Weyl’smathematical curiosity was not restricted to integral equations. In his own words, hewas captivated by all of Hilbert’s mathematics. Later he wrote:8

I resolved to study whatever this man [Hilbert] had written. At the endof my first year I went home with the “Zahlbericht” under my arm,and during the summer vacation I worked my way through it – withoutany previous knowledge of elementary number theory or Galois theory.These were the happiest months of my life, whose shine, across yearsburdened with our common share of doubt and failure, still comforts mysoul.

We see that Weyl in Göttingen was exposed to and responded to the new andexciting ideas which were sprouting in the mathematical world at the time. Hismathematical education was strongly influenced by his advisor Hilbert.

7For additional material see also Tollmien’s web page: http://www.tollmien.com/.8Cited from the Weyl article in “MacTutor History of Mathematics Archive”.

http://www.tollmien.com/


On the other side, Noether lived in the small and quiet mathematical world of Er-langen. Her thesis, supervised by Paul Gordan, belongs to classical invariant theory,in the framework of so-called symbolic computations. Certainly this did no longerbelong to the main problems which dominated mathematical research in the begin-ning of the 20th century. It is a well-known story that after Hilbert in 1888 had provedthe finiteness theorem of invariant theory which Gordan had unsuccessfully tried fora long time, then Gordan did not accept Hilbert’s existence proof since that was notconstructive in his (Gordan’s) sense. He declared that Hilbert’s proof was “theology,not mathematics”. Emmy Noether’s work was fully integrated into Gordan’s formal-ism and so, in this way, she was not coming near to the new mathematical ideas ofthe time.9 In later years she described the work of her thesis as rubbish (“Mist” inGerman10). In a letter of April 14, 1932 to Hasse she wrote:

Ich habe das symbolische Rechnen mit Stumpf und Stil verlernt.

I have completely forgotten the symbolic calculus.

We do not know when Noether had first felt the desire to update her mathe-matical background. Maybe the discussions with her father helped to find her way;he corresponded with Felix Klein in Göttingen and so was well informed about themathematical news from there. She herself reports that it was mainly Ernst Fischerwho introduced her to what was then considered “modern” mathematics. Fischercame to Erlangen in 1911, as the successor of the retired Gordan.11 In her curriculumvitae which she submitted in 1919 to the Göttingen Faculty on the occasion of herHabilitation, Noether wrote:

Wissenschaftliche Anregung verdanke ich wesentlich dem persönlichenmathematischen Verkehr in Erlangen und in Göttingen. Vor allem binich Herrn E. Fischer zu Dank verpflichtet, der mir den entscheidendenAnstoß zu der Beschäftigung mit abstrakter Algebra in arithmetischerAuffassung gab, was für all meine späteren Arbeiten bestimmend blieb.

I obtained scientific guidance and stimulation mainly through personalmathematical contacts in Erlangen and in Göttingen. Above all I amindebted to Mr. E. Fischer from whom I received the decisive impulse tostudy abstract algebra from an arithmetical viewpoint, and this remainedthe governing idea for all my later work.

9Well, Noether had studied one semester in Göttingen, winter 1903/04. But she fell ill during that time andhad to return to her home in Erlangen, as Tollmien [Tol90] reports. We did not find any indication that thisparticular semester has had a decisive influence on her mathematical education.

10Cited from Auguste Dick’s Noether biography [Dic70].11More precisely: Gordan retired in 1910 and was followed by Erhard Schmidt who, however, left Erlangen

one year later already and was followed in turn by Ernst Fischer. – The name of Fischer is known from theFischer–Riesz theorem in functional analysis.


Thus it was Fischer under whose direction Emmy Noether’s mathematical outlookunderwent the “transition from Gordan’s formal standpoint to the Hilbert method ofapproach”, as Weyl stated in [Wey35].

We may assume that Emmy Noether studied, like Weyl, all of Hilbert’s papers, atleast those which were concerned with algebra or arithmetic. In particular she wouldhave read the paper [Hil90] where Hilbert proved that every ideal in a polynomialring is finitely generated; in her famous later paper [Noe21] she considered arbitraryrings with this property, which today are called “Noetherian rings”. We may alsoassume that Hilbert’s Zahlbericht too was among the papers which Emmy Noetherstudied; it was the standard text which every young mathematician of that time readif he/she wished to learn algebraic number theory. We know from a later statementthat she was well acquainted with it – although at that later time she rated it rathercritically12, in contrast to Weyl who, as we have seen above, was enthusiastic aboutit. But not only Hilbert’s papers were on her agenda; certainly she read Steinitz’greatpaper Algebraische Theorie der Körper (Algebraic Theory of Fields) [Ste10] whichmarks the start of abstract field theory. This paper is often mentioned in her laterpublications, as the basis for her abstract viewpoint of algebra.

4.3.2 Meeting in Göttingen 1913

Hermann Weyl says in [Wey35], referring to the year 1913:

… She must have been to Göttingen about that time, too, but I supposeonly on a visit with her brother Fritz. At least I remember him muchbetter than her from my time as a Göttinger Privatdozent, 1910–1913.

We may conclude that he had met Emmy Noether in Göttingen about 1913, but alsothat she did not leave a lasting impression on him on that occasion.

As Tollmien [Tol90] reports, it was indeed 1913 when Emmy Noether visitedGöttingen for a longer time (together with her father Max Noether). Although wehave no direct confirmation we may well assume that she met Weyl during this time.In the summer semester 1913 Weyl gave two talks in the Göttinger MathematischeGesellschaft. In one session he reported on his new book Die Idee der RiemannschenFläche (The idea of the Riemann surface) [Wey13], and in another he presented hisproof on the equidistribution of point sequences modulo 1 in arbitrary dimensions[Wey16] – both pieces of work have received the status of a “classic” by now. Cer-tainly, Max Noether as a friend of Klein will have been invited to the sessions ofthe Mathematische Gesellschaft, and his daughter Emmy with him. Before and afterthe session people would gather for discussion, and from all we know about EmmyNoether she would not have hesitated to participate in the discussions. From what

12In a letter of November 17, 1926 to Hasse; see [LR06]. Olga Taussky-Todd [Tau81] reports from later timein Bryn Mawr that once “Emmy burst out against the Zahlbericht, quoting also Artin as having said that it delayedthe development of algebraic number theory by decades”.


we have said in the foregoing section we can conclude that her mathematical statuswas up-to-date and well comparable to Weyl’s, at least with respect to algebra andnumber theory.

Unfortunately we do not know anything about the possible subjects of the dis-cussions of Emmy Noether with Weyl. It is intriguing to think that they could havetalked about Weyl’s new book The idea of the Riemann surface. Weyl in his book de-fines a Riemann surface axiomatically by structural properties, namely as a connectedmanifold X with a complex 1-dimensional structure. This was a completely new ap-proach, a structural viewpoint. Noether in her later period used to emphasize on everyoccasion the structural viewpoint. The structure in Weyl’s book is an analytic one,and he constructs an algebraic structure from this, namely the field of meromorphicfuntions, using the so-called Dirichlet principle – whereas Emmy Noether in her laterpapers always starts from the function field as an algebraic structure. See for instanceher report [Noe19]. There she did not cite Weyl’s book but, of course, this does notmean that she did not know it.

We observe that the starting idea in Weyl’s book was the definition and use ofan axiomatically defined topological space13. We wonder whether this book was thefirst instance where Emmy Noether was confronted with the axioms of what later wascalled a topological space. It is not without reason to speculate that her interest intopology was inspired by Weyl’s book. In any case, from her later cooperation withPaul Alexandrov we know that she was acquainted with problems of topology; hercontribution to algebraic topology was the notion of Betti group instead of the Bettinumber which was used before. Let us cite Alexandrov in his autobiography [Ale80]:

In the middle of December Emmy Noether came to spend a month inBlaricum. This was a brilliant addition to the group of mathematiciansaround Brouwer. I remember a dinner at Brouwer’s in her honour duringwhich she explained the definition of the Betti groups of complexes,which spread around quickly and completely transformed the whole oftopology.

This refers to December 1925. Blaricum was the place where L. E. J. Brouwer lived.We have mentioned this contact of Emmy Noether to the group around Brouwer

since Weyl too did have mathematical contact with Brouwer. In fact, in his book Theidea of the Riemann surface Weyl mentioned Brouwer as a source of inspiration. Hewrites:

In viel höherem Maße, als aus den Zitaten hervorgeht, bin ich dabei durchdie in den letzten Jahren erschienenen grundlegenden topologischen Un-tersuchungen Brouwers, deren gedankliche Schärfe und Konzentrationman bewundern muß, gefördert worden; …

13The Hausdorff axiom was not present in the first edition. This gap was filled in later editions.


I have been stimulated – much more than the citations indicate – by therecent basic topological investigations of Brouwer, whose ideas have tobe admired in their sharpness and concentration;…

Brouwer’s biographer van Dalen reports that Weyl and Brouwer met several times inthe early 1920s [vD99]. By the way, Emmy Noether, HermannWeyl and L. E. J. Brou-wer met in September 1920 in Bad Nauheim, at the meeting of the DMV.14

Returning to the year 1913 in Göttingen: In the session of July 30, 1913 of the Göt-tinger Mathematische Gesellschaft, Th. v. Kármán reported on problems connectedwith a recent paper on turbulence by Emmy Noether’s brother Fritz. Perhaps Fritz toowas present in Göttingen on this occasion, and maybe this was the incident why Weylhad remembered not only Emmy but also Fritz? That he remembered Fritz “muchbetter” may be explained by the topic of Fritz’ paper; questions of turbulence lead toproblems about partial differential equations, which was at that time more close toWeyl’s interests than were algebraic problems which Emmy pursued.

4.4 The second period: 1915–1920

In these years Emmy Noether completed several papers which are of algebraic na-ture, mostly about invariants, inspired by the Göttingen mathematical atmospheredominated by Hilbert. She also wrote a report in the Jahresbericht der DMV onalgebraic function fields, in which Noether compares the various viewpoints of thetheory: analytic, geometric and algebraic (which she called “arithmetic”) and shepoints out the analogies to the theory of number fields. That was quite well knownto the people working with algebraic functions, but perhaps not written up system-atically as Emmy Noether did. Generally, these papers of hers can be rated as goodwork, considering the state of mathematics of the time, but not as outstanding. It isunlikely that Hermann Weyl was particularly interested in these papers; perhaps hedidn’t even know about them.15

But this would change completely with the appearance of Noether’s paper oninvariant variation problems [Noe18] (Invariante Variationsprobleme). The mainresult of this paper is of fundamental importance in many branches of theoreticalphysics even today. It shows a connection between conservation laws in physics andthe symmetries of the theory. It is probably the most cited paper of Emmy Noetherup to the present day. In 1971 an English translation appeared [Tav71], and in 2004a French translation with many comments [KS04].

In 1918, when the paper appeared, its main importance was seen in its applicabilityin the framework of Einstein’s relativity theory. Einstein wrote to Hilbert in a letterof May 24, 1918:

14DMV = Deutsche Mathematiker Vereinigung = German Mathematical Society.15In those years Weyl was no more in Göttingen but held a professorship at ETH in Zürich.


Gestern erhielt ich von Frl. Noether eine sehr interessante Arbeit überInvariantenbildung. Es imponiert mir, dass man diese Dinge von so all-gemeinem Standpunkt übersehen kann … Sie scheint ihr Handwerk zuverstehen.

Yesterday I received from Miss Noether a very interesting paper on theformation of invariants. I am impressed that one can handle those thingsfrom such a general viewpoint … She seems to understand her job.

Einstein had probably met Emmy Noether already in 1915 during his visit to Göttin-gen.

Emmy Noether’s result was the fruit of a close cooperation with Hilbert and withKlein in Göttingen during the past years. As Weyl [Wey35] reports:

Hilbert at that time was over head and ears in the general theory ofrelativity, and for Klein, too, the theory of relativity brought the lastflareup of his mathematical interest and production.

Emmy Noether, although she was doubtless influenced, not only assisted them but herwork was a genuine production of her own. In particular, the connection of invariantswith the symmetry groups, with its obvious reference to Klein’s Erlanger program,caught the attention of the world of mathematicians and theoretical physicists. 16

Noether’s work in this direction has been described in detail in, e.g., [Row99], [KS04],[Wue05].

It is inconceivable that Hermann Weyl did not take notice of this important workof Emmy Noether. At that time Weyl, who was in correspondence with Hilbert andEinstein, was also actively interested in the theory of relativity; his famous bookRaum, Zeit, Materie (Space, Time, Matter) had just appeared. Emmy Noether hadcitedWeyl’s book17, and almost certainly she had sent him a reprint of her paper. Thus,through the medium of relativity theory there arose mathematical contact betweenthem.18 Although we do not know, it is well conceivable that there was an exchangeof letters concerning the mathematical theory of relativity. From now on Weyl wouldnever remember her brother Fritz better than Emmy.

In 1919 Emmy Noether finally got her Habilitation. Already in 1915 Hilbert andKlein, convinced of her outstanding qualification, had recommended her to apply forHabilitation. She did so, but it is a sad story that it was unsuccessful because of hergender although her scientific standing was considered sufficient. The incident is toldin detail in Tollmien’s paper [Tol90]. Thus her Habilitation was delayed until 1919after the political and social conditions had changed.

16In [Row99] it is said that nevertheless “few mathematicians and even fewer physisists ever read Noether’soriginal article …”.

17The citation is somewhat indirect. Noether referred to the literature cited in a paper by Felix Klein [Kle18],and there we find Weyl’s book mentioned. In a second paper of Noether [Noe23] Weyl’s book is cited directly.

18Added in proof: We read in [Row99] that there is a reference to Noether in Weyl’s book, tucked away in afootnote.


We see again the difference between the scientific careers of Weyl and of EmmyNoether. Weyl had his Habilitation already in 1910, and since 1913 he held a pro-fessorship in Zürich. Emmy Noether’s Habilitation was possible only nine yearslater than Weyl’s. As is well known, she never in her life got a permanent position;although in the course of time she rose to become one of the leading mathematiciansin the world.

4.5 The third period: 1920–1932

The third period of Noether’s mathematical life starts with the great paper Idealtheoriein Ringbereichen (Ideal theory in rings) [Noe21].19 After Hilbert had shown in 1890that in a polynomial ring (over a field as base) every ideal is finitely generated, Noethernow takes this property as an axiom and investigates the primary decomposition ofideals in arbitrary rings satisfying this axiom. And she reformulates this axiom as an“ascending chain condition” for ideals. Nowadays such rings are called Noetherian.The paper appeared in 1921.

We note that she was nearly 40 years old at that time. The mathematical life ofEmmy Noether is one of the counterexamples to the dictum that mathematics is ascience for the young and the most creative work is done before 40. Emmy Noetherwould not have been a candidate for the Fields Medal if it had already existed at thattime.

4.5.1 Innsbruck 1924 and the method of abstraction

We do not know whether and how Weyl took notice of the above-mentioned paperof Noether [Noe21]. But her next great result, namely the follow-up paper [Noe26]on the ideal theory of what are now called Dedekind rings, was duly appreciated byWeyl. At the annual DMV-meeting in 1924 in Innsbruck Noether reported about it[Noe24]. And Weyl was chairing that session; so we know that he was informed firsthand about her fundamental results.

In her talk, Emmy Noether defined Dedekind rings by axioms and showed thatevery ring satisfying those axioms admits a unique factorization of ideals into primeideals. Well, Noether did not use the terminology “Dedekind ring”; this name wascoined later. Instead, she used the name “5-axioms-ring” since in her enumerationthere were 5 axioms. Then she proved that the ring of integers in a number fieldsatisfies those axioms, and similarly in the funtion field case. This is a good exampleof Noether’s “method of abstraction”. By working solely with those axioms she firstgeneralized the problem, and it turned out that by working in this generalization the

19Sometimes the earlier investigation jointly with Schmeidler [NS20] is also counted as belonging to thisperiod.


proof of prime decomposition is simplified if compared with the former proofs (twoof which had been given by Hilbert [Hil94]).

How did Weyl react to Noether’s method of abstraction? At that time, this methodmet sometimes with skepticism and even rejection by mathematicians. But Hilbert invarious situations had already taken first steps in this direction and so Weyl, havingbeen Hilbert’s doctorand, was not against Noether’s method. After all, in his bookSpace, Time, Matter Weyl had introduced vector spaces by axioms, not as n-tuples20.

Weyl’s reaction can be extracted implicitly from an exchange of letters with Hassewhich happened seven years later. The letter of Weyl is dated December 8, 1931.At that time Weyl held a professorship in Göttingen (since 1930) as the successor ofHilbert. Thus Emmy Noether was now his colleague in Göttingen. Hasse at that timeheld a professorship in Marburg (also since 1930) as the successor of Hensel. Theoccasion ofWeyl’s letter was the theorem that every simple algebra over a number fieldis cyclic; this had been established some weeks ago by Brauer, Hasse and Noether,and the latter had informed Weyl about it. So Weyl congratulated Hasse for thissplendid achievement. And he recalled the meeting in Innsbruck 1924 when he firsthad met Hasse.

For us, Hasse’s reply to Weyl’s letter is of interest.21 Hasse answered on De-cember 15, 1931. First he thanked Weyl for his congratulations, but at the sametime pointed out that the success was very essentially due also to the elegant the-ory of Emmy Noether, as well as the p-adic theory of Hensel. He also mentionedMinkowski in whose work the idea of the Local-Global Principle was brought to lightvery clearly. And then Hasse continued, recalling Innsbruck:

Auch ich erinnere mich sehr gut an Ihre ersten Worte zu mir anläßlichmeines Vortrages über die erste explizite Reziprozitätsformel für höherenExponenten in Innsbruck. Sie zweifelten damals ein wenig an der innerenBerechtigung solcher Untersuchungen, indem Sie ins Feld führten, es seidoch gerade Hilberts Verdienst, die Theorie des Reziprozitätsgesetzesvon den expliziten Rechnungen früherer Forscher, insbesondere Kum-mers, befreit zu haben.

I too remember very well your first words to me on the occasion of mytalk in Innsbruck, about the first explicit reciprocity formula for higherexponent. You somewhat doubted the inner justification of such inves-tigations, by pointing out that Hilbert had freed the theory of the reci-procity law from the explicit computations of former mathematicians,in particular Kummer’s.

We conclude: Hasse in Innsbruck had talked on explicit formulas and Weyl hadcritized this, pointing out that Hilbert had embedded the reciprocity laws into more

20This has been expressly remarked by Mac Lane [ML81].21We have found Hasse’s letter in the Weyl legacy in the archive of the ETH in Zürich.


structural results. Probably Weyl had in mind the product formula for the so-calledHilbert symbol which, in a sense, comprises all explicit reciprocity formulas.22 Usu-ally this product formula was considered as the final word on reciprocity, and Weyltoo adhered to this opinion. But for Hasse this was only the starting point for deriv-ing explicit, constructive reciprocity formulas, using heavily (if possible) the p-adicmethods of Hensel.

We can fairly well reconstruct the situation in Innsbruck: Emmy Noether’s talkhad been very abstract, and Hasse’s achievement was in some sense the oppositesince he was bent on explicit formulas, and quite involved ones too.23 Weyl had beenimpressed by Emmy Noether’s achievements which he considered as continuing alongthe lines set by Hilbert’s early papers on number theory. In contrast, he consideredHasse’s work as pointing not to the future but to the mathematical past.

We have mentioned here these letters Weyl–Hasse in order to put into evidencethat already in 1924, Weyl must have had a very positive opinion on Emmy Noether’smethods, even to the point of preferring it to explicit formulas.

But as it turned out, Hasse too had been impressed by Noether’s lecture. In thecourse of the years after 1924, as witnessed by the Hasse–Noether correspondence[LR06], Hasse became more and more convinced about Noether’s abstract methodswhich, in his opinion, served to clarify the situation; he used the word “durchsichtig”(lucid). Hasse’s address at the DMV meeting in Prague 1929 [Has30b] expresses hisviews very clearly. Hensel’s p-adic methods could also be put on an abstract base,due to the advances in the theory of valuations.24 But on the other hand, Hasse wasnever satisfied with abstract theorems only. In his cited letter to Weyl 1931 he referredto his (Hasse’s) class field report Part II [Has30a] which had appeared just one yearearlier. There, he had put Artin’s general reciprocity law25 as the base, and from thisstructural theorem he was able to derive all the known reciprocity formulas. Hasseclosed his letter with the following:

… Ich kann aber natürlich gut verstehen, daß Dinge wie diese explizi-ten Reziprozitätsformeln einem Manne Ihrer hohen Geistes- und Ge-schmacksrichtung weniger zusagen, als mir, der ich durch die abstrakteMathematik Dedekind–E. Noetherscher Art nie restlos befriedigt bin,ehe ich nicht zum mindesten auch eine explizite, formelmäßige kon-struktive Behandlung daneben halten kann. Erst von der letzteren könnensich die eleganten Methoden und schönen Ideen der ersteren wirklichvorteilhaft abheben.

22But Hilbert was not yet able to establish his product formula in full generality. We refer to the beautiful andcomplete treatment in Hasse’s class field report, Part 2 [Has30a] which also contains the most significant historicreferences.

23Hasse’s Innsbruck talk is published in [Has25]. The details are found in volume 154 of Crelle’s Journalwhere Hasse had published five papers on explicit reciprocity laws.

24For this see [Roq02]. See also Chapter 9, in particular page 232.25By the way, this was the first treatment of Artin’s reciprocity law in book form after Artin’s original paper

1927. We refer to our forthcoming book on the Artin–Hasse correspondence.


… But of course I well realize that those explicit reciprocity formulasmay be less attractive to a man like you with your high mental powers andtaste, as to myself. I am never fully satisfied by the abstract mathematicsof Dedekind–E. Noether type before I can also supplement it by at leastone explicit, computational and constructive treatment. It is only incomparison with the latter that the elegant methods and beautiful ideasof the former can be appreciated advantageously.

Here, Hasse touches a problem which always comes up when, as Emmy Noetherpropagated, the abstract methods are put into the foreground. Namely, abstractionand axiomatization is not to be considered as an end in itself but it is a method to dealwith concrete problems of substance. But Hasse was wrong when he supposed thatWeyl did not see that problem. Even in 1931, the same year as the above cited letters,Weyl gave a talk on abstract algebra and topology as two ways of mathematicalcomprehension [Wey32]. In this talk Weyl stressed the fact that axiomatization isnot only a way of securing the logical truth of mathematical results, but that it hadbecome a powerful tool of concrete mathematical research itself, in particular underthe influence of Emmy Noether. But he also said that abstraction and generalizationdo not make sense without mathematical substance behind it. This is close to Hasse’sopinion as expressed in his letter above.26 The mathematical work of both Weyl andHasse puts their opinions into evidence.

At the same conference [Wey32] Weyl also said that the “fertility of these abstract-ing methods is approaching exhaustion”. This, however, met with sharp protests byEmmy Noether, as Weyl reports in [Wey35]. In fact, today most of us would agreewith Noether. The method of abstracting and axiomatizing has become a natural andpowerful tool for the mathematician, with striking successes until today. In Weyl’sletter to Hasse (which we have not cited fully) there are passages which seem toindicate that in principle he (Weyl) too would agree with Emmy Noether. For, heencourages Hasse to continue his work in the same fashion, and there is no mention ofan impending “exhaustion”. He closes his letter with the following sentence which,in our opinion, shows his (Weyl’s) opinion of how to work in mathematics:

Es freut mich besonders, daß bei Ihnen die in Einzelleistungen sich be-währende wissenschaftliche Durchschlagskraft sich mit geistigem Weit-blick paart, der über das eigene Fach hinausgeht.

In particular I am glad that your scientific power, tested in various specialaccomplishments, goes along with a broad view stretching beyond yourown special field.

26Even more clearly Hasse has expressed his view in the foreword to his beautiful and significant book onabelian fields [Has52].


4.5.2 Representations: 1926/27

We have made a great leap from 1924 to 1931. Now let us return and proceed alongthe course of time. In the winter semester 1926/27 Hermann Weyl stayed in Göttingenas a visiting professor, and he lectured on representations of continuous groups. In[Wey35] he reports:

I have a vivid recollection of her [Emmy Noether] … She was in the au-dience; for just at that time the hypercomplex number systems and theirrepresentations had caught her interest and I remember many discussionswhen I walked home after the lectures, with her and von Neumann, whowas in Göttingen as a Rockefeller Fellow, through the cold, dirty, rain-wet streets of Göttingen.

This gives us information not only about the weather conditions in Göttingen in wintertime but also that a lively discussion between Weyl and Emmy Noether had developed.

We do not know precisely when Emmy Noether first had become interested inthe representation theory of groups and algebras, or “hypercomplex systems” in herterminology. In any case, during the winter semester 1924/25 in Göttingen she hadgiven a course on the subject. And in September 1925 she had talked at the annualmeeting of the DMV in Danzig on “Group characters and ideal theory”. There sheadvocated that the whole representation theory of groups should be subsumed underthe theory of algebras and their ideals. She showed how the Wedderburn theoremsfor algebras are to be interpreted in representation theory, and that the whole theoryof Frobenius on group characters is subsumed in this way. Although she announceda more detailed presentation in the Mathematische Annalen, the mathematical publichad to wait until 1929 for the actual publication [Noe29]27. Noether was not a quickwriter; she developed her ideas again and again in discussions, mostly on her walkswith students and colleagues into the woods around Göttingen, and in her lectures.

The text of her paper [Noe29] consists essentially of the notes taken by van derWaerden at her lecture in the winter semester 1927/28. Although the main motivationof Noether was the treatment of Frobenius’ theory of representations of finite groups,it turned out that finite groups are treated on the last two pages only – out of a totalof 52 pages. The main part of the paper is devoted to introducing and investigatinggeneral abstract notions, capable of dealing not only with the classical theory of finitegroup representations but with much more. Again we see the power of Noether’sabstracting methods. The paper has been said to constitute “one of the pillars ofmodern linear algebra”.28

We can imagine Emmy Noether in her discussions with Weyl on the cold, wetstreets in Göttingen 1926/27, explaining to him the essential ideas which were to be-come the foundation of her results in her forthcoming paper [Noe29]. We do not know

27This appeared in the Mathematische Zeitschrift and not in the Annalen as announced by Noether in Danzig.28Cited from [Cur99] who in turn refers to Bourbaki.


to which extent these ideas entered Weyl’s book [Wey39] on classical groups. Afterall, the classical groups which are treated in Weyl’s book are infinite while Noether’stheory aimed at the representation of finite groups. Accordingly, in Noether’s workthere appeared a finiteness condition for the algebras considered, namely the descend-ing chain condition for (right) ideals. If one wishes to use Noether’s results for infinitegroups one first has to generalize her theory such as to remain valid in more generalcases too. Such a generalization did not appear until 1945; it was authored by NathanJacobson [Jac45]. He generalized Noether’s theory to simple algebras containing atleast one irreducible right ideal.

At this point let me tell a story which I witnessed in 1947. I was a young studentin Hamburg then. In one of the colloquium talks the speaker was F. K. Schmidtwho recently had returned from a visit to the USA, and he reported on a new paperby Jacobson which he had discovered there. This was the above mentioned paper[Jac45].29 F. K. Schmidt was a brilliant lecturer and the audience was duly impressed.In the ensuing discussion Ernst Witt, who was in the audience, commented that allthis had essentially been known to Emmy Noether already.

Witt did not elaborate on his comment. But he had been one of the “Noetherboys” in 1932/33, and so he had frequently met her. There is no reason to doubt hisstatement. It may well have been that she had told him, and perhaps others too, thather theory could be generalized in the sense which later had been found by Jacobson.Maybe she had just given a hint in this direction, without details, as was her usualcustom. In fact, reading Noether’s paper [Noe29] the generalization is obvious toany reader who is looking for it.30 It is fascinating to think that the idea for sucha generalization arose from her discussions with Weyl in Göttingen in 1927, wheninfinite groups were discussed and the need to generalize her theory became apparent.

By the way, Jacobson and Emmy Noether had met in 1934 in Princeton, when shewas running a weekly seminar. We cannot exclude the possibility that she had givena hint to him too, either in her seminar or in personal discussions. After all, this washer usual style, as reported by van der Waerden [vdW35].

4.5.3 A letter from N to W: 1927

As stated in the introduction we have not found letters from Emmy Noether to Weyl,with one exception. That exception is kept in the archive of the ETH in Zürich. It iswritten by Emmy Noether and dated March 12, 1927. This is shortly after the endof the winter semester 1926/27 when Weyl had been in Göttingen as reported in theprevious section. Now Weyl was back in Zürich and they had to write letters insteadof just talking.

29In those post-war years it was not easy to get hold of books or journals from foreign countries, and so the1945 volume of the “Transactions of AMS” was not yet available at the Hamburg library.

30A particularly short and beautiful presentation is to be found in Artin’s article [Art50] where he refers toTate.


The letter concerns PaulAlexandrov and Heinz Hopf and their plan to visit Prince-ton in the academic year 1927/28.

We have already mentioned Alexandrov in Section 4.3.2 in connection withNoether’s contributions to topology. From 1924 to 1932 he spent every summerin Göttingen, and there developed a kind of friendly relationship between him andEmmy Noether. The relation of Noether to her “Noether boys” has been described byAndré Weil as like a mother hen to her fledglings [Wei93]. Thus Paul Alexandrov wasaccepted by Emmy Noether as one of her fledglings. In the summer semester 1926Heinz Hopf arrived in Göttingen as a postdoc from Berlin and he too was acceptedas a fledgling. Both Alexandrov and Hopf became close friends and they decided totry to go to Princeton University in the academic year 1927/28.

Perhaps Emmy Noether had suggested this; in any case she helped them to obtaina Rockefeller grant for this purpose. It seems that Weyl also had lent a helping hand,for in her letter to him she wrote:

… Jedenfalls danke ich Ihnen sehr für Ihre Bemühungen; auch Alexan-drov und Hopf werden Ihnen sehr dankbar sein und es scheint mir sicher,dass wenn die formalen Schwierigkeiten erst einmal überwunden sind,Ihr Brief dann von wesentlichem Einfluss sein wird.

… In any case I would like to thank you for your help; Alexandrov andHopf too will be very grateful to you. And I am sure that if the formalobstacles will be overcome then your letter will be of essential influence.

The “formal obstacles” which Noether mentioned were, firstly, the fact that orig-inally the applicants (Alexandrov and Hopf) wished to stay for a period less than anacademic year in Princeton (which later they extended to a full academic year), andsecondly, that Hopf’s knowledge of the English language seemed not to be sufficientin the eyes of the Rockefeller Foundation (but Noether assured them that Hopf wantedto learn English).31 But she mentioned there had been letters sent to Lefschetz andBirkhoff and that at least the latter had promised to approach the Rockefeller Foun-dation to make an exception.

Alexandrov was in Moscow and Hopf in Berlin at the time, and so the mother henacted as representative of her two chickens.32

About Alexandrov’s and Hopf’s year in Princeton we read in the Alexandrovarticle of MacTutor’s History of Mathematics archive:

Alexandrov and Hopf spent the academic year 1927–28 at Princeton inthe United States. This was an important year in the development of

31It seems that his knowledge of English had improved in the course of time since Heinz Hopf had been electedpresident of the IMU (International Mathematical Union) in 1955 till 1958.

32 In Kimberling’s article [Kim81] it is reported: “Handwritten letters dated 6/1/27 and 7/3/27 from EmmyNoether to W. W. Tinsdale supporting Hopf’s application are preserved in the International Educational BoardCollection at the Rockefeller Archive Center.” Probably the letters from Weyl too are preserved there but wehave not checked this.


topology with Alexandrov and Hopf in Princeton and able to collaboratewith Lefschetz, Veblen and Alexander.

The letter from Noether to Weyl shows that both N. and W. were instrumental inarranging this important Princeton year for Alexandrov and Hopf. Both were alwaysready to help young mathematicians to find their way.

Remark. Later in 1931, when Weyl had left Zürich for Göttingen, it was Heinz Hopfwho succeeded Weyl in the ETH Zürich. At those times it was not uncommon that theleaving professor would be asked for nominations if the faculty wished to continuehis line. We can well imagine that Weyl, who originally would have preferred Artin33, finally nominated Heinz Hopf for this position. If so then he would have discussedit with Emmy Noether since she knew Hopf quite well. It may even have been thatshe had taken the initiative and proposed to Weyl the nomination of Hopf. In fact, inthe case of Alexandrov she did so in a letter to Hasse dated October 7, 1929 when itwas clear that Hasse would change from Halle to Marburg. There she asked Hassewhether he would propose the name ofAlexandrov as a candidate in Halle.34 It seemsrealistic to assume that in the case of Heinz Hopf she acted similarly.

Remark. The above mentioned letter of Noether to Weyl contains a postscript whichgives us a glimpse of the mathematical discussion between the two (and it is the onlywritten document for this). It reads:

Die Mertens-Arbeit, von der ich Ihnen sprach, steht Monatshefte, Bd. 4.Ich dachte an den Schluss, Seite 329. Es handelt sich hier aber doch nurum Determinanten-Relationen, sodass es für Sie wohl kaum in Betrachtkommt.

The Mertens paper which I mentioned to you is contained in volume 4of the Monatshefte. I had in mind the end of the paper, page 329. Butthis is concerned with determinant relations only, hence it will perhapsnot be relevant to your purpose.

That Mertens paper is [Mer93]. We have checked the cited page but did not findany hint which would connect to Weyl’s work. Perhaps someone else will be able tointerpret Noether’s remark.

4.5.4 Weyl in Göttingen: 1930–1933

Weyl in [Wey35] reports:

When I was called permanently to Göttingen in 1930, I earnestly tried toobtain from the Ministerium a better position for her [Emmy Noether],

33In fact, in 1930 Artin received an offer from the ETH Zürich which, however, he finally rejected.34This however, did not work out. The successor of Hasse in Halle was Heinrich Brandt, known for the

introduction of “Brandt’s gruppoid” for divisor classes in simple algebras over number fields.


because I was ashamed to occupy such a preferred position beside herwhom I knew my superior as a mathematician in many respects.

We see that by now, Weyl was completely convinced about the mathematicalstature of Emmy Noether. After all, Emmy Noether in 1930 was the world-wideacknowledged leader of abstract algebra, and her presence in Göttingen was the mainattraction for young mathematicians from all over the world to visit the MathematicalInstitute and study with her.

E. Noether and H. Weyl (with hat) among a group of mathematicians

It would be interesting to try to find out which “better position” Weyl had in mindin his negotiations with the Ministerium in Berlin. Maybe he wished tenure for her,and an increase of her salary. The archives in Berlin will perhaps have the papersand reports of Weyl’s negotiations. From those papers one may be able to extractthe reasons for the rejection. But the opposition against Noether’s promotion did notonly come from the Ministerium in Berlin. It seems that a strong opposition camealso from among the mathematician colleagues in Göttingen, for Weyl continues withhis report as follows:

… nor did an attempt [succeed] to push through her election as a memberof the Göttinger Gesellschaft der Wissenschaften. Tradition, prejudice,external considerations, weighted the balance against her scientific mer-its and scientific greatness, by that time denied by no one.

I do not know whether there exist minutes of the meetings of theGöttingerGesellschaftder Wissenschaften in 1930. If so then it would be interesting to know the tradi-tional, biased and external arguments which Weyl said were put forward against


Emmy Noether from the members of the Gesellschaft der Wissenschaften. Was itstill mainly her gender? Or was it the opposition to her “abstract” mathematicalmethods? In any case, the decision not to admit Emmy Noether as a member of theGöttinger Gesellschaft der Wissenschaften is to be regarded as an injustice to her anda lack of understanding of the development of modern mathematics. After all, theEmmy Noether of 1930 was quite different from the Emmy Noether of 1915. Nowin 1930, she had already gone a long way “on her own completely original mathe-matical path” , and her “working and conceptual methods had spread everywhere”.She could muster high-ranking colleagues and students who were fascinated by herway of mathematical thinking.

Nevertheless it seems that in Göttingen, even among mathematicians, there existedsome opposition against her abstract methods. Olga Taussky-Todd recalls in [Tau81]her impression of the Göttingen mathematical scene:

… not everybody liked her [Emmy Noether], and not everybody trustedthat her achievements were what they later were accepted to be.

One day Olga Taussky had been present when one of the senior professors talkedvery roughly to Emmy Noether. (Later he apologized to her for this insult.) WhenEmmy Noether had her 50th birthday in 1932 then, as Olga Taussky recalls, nobodyat Göttingen had taken notice of it, although at that time all birthdays were publishedin the Jahresbericht of the DMV. 35 Reading all this, I can understand Emmy Noetherwhen later in 1935 she said to Veblen about her time in the USA:

The last year and a half had been the very happiest in her whole life, forshe was appreciated in Bryn Mawr and Princeton as she had never beenappreciated in her own country.36

Thus it seems that Weyl’s statement that “her scientific merits and scientific great-ness by that time was denied by no one” did not describe the situation exactly. Perhaps,since Weyl was the “premier professor” of mathematics at Göttingen, and since hewas known to respect and acknowledge Noether’s merits and scientific greatness,nobody dared to tell him if he disagreed. Olga Taussky-Todd remembers that

… outside of Göttingen, Emmy was greatly appreciated in her country.

We may add that this was not only so in her country but also world-wide. Andof course also in Göttingen there was an ever-growing fraction of mathematicians,including Weyl, who held Noether in high esteem.

As to HermannWeyl, let us cite Saunders Mac Lane who was a student at Göttingenin the period 1931–33. We read in [Mac81]:

35But Hasse in Marburg had sent her a birthday cake, together with a paper which he had dedicated to her.The paper was [Has33a]. See [LR06].

36Cited from a letter of Abraham Flexner to President Park of Bryn Mawr; see [Roq07b].


When I first came to Göttingen I spoke to Professor Weyl and expressedmy interest in logic and algebra. He immediately remarked that in al-gebra Göttingen was excellently represented by Professor Noether; herecommended that I attend her courses and seminars … By the time ofmy arrival she was Ausserordentlicher Professor. However, it was clearthat in the view of Weyl, Hilbert, and the others, she was right on thelevel of any of the full professors. Her work was much admired and herinfluence was widespread.

Mac Lane sometimes joined the hiking parties (“Ausflug”) of Emmy Noetherand her class to the hills around Göttingen. Noether used these hiking parties todiscuss “algebra, other mathematical topics and Russia”.37 It seems that Weyl toojoined those excursions occasionally. There is a nice photo of Noether with Weyland family, together with a group of mathematicians posing in front of the “GasthofVollbrecht”. The photo is published in [BS81] and dated 1932. Since Artin is seen asa member of the hiking party, it seems very probable that the photo was taken on theoccasion of Artin’s famous Göttingen lectures on class field theory which took placefrom February 29 to March 2, 1932.38 This was a big affair and a number of peoplecame from various places in order to listen to Artin lecturing on the new face of classfield theory. The lectures were organized by Emmy Noether. Since she was not afull professor and, accordingly, had no personal funds to organize such meetings wesuppose that one of her colleagues, probably Weyl, had made available the necessaryfinancial means for this occasion. In any case we see that by now she was able toget support for her activities in Göttingen, not only for the Artin lectures but also forother speakers.

The International Congress of Mathematicians took place in September 1932in Zürich. Emmy Noether was invited to deliver one of the main lectures there.Usually, proposals for invited speakers at the IMU conferences were submitted bythe presidents of the national mathematical organizations which were members ofthe IMU. In 1931/32 Hermann Weyl was president (“Vorsitzender”) of the DMV.So it appears that Weyl had his hand in the affair when it came to proposing EmmyNoether as a speaker from Germany. The proposal had to be accepted by the executivecommittee. The nomination of Emmy Noether was accepted and this shows the greatrespect and admiration which Emmy Noether enjoyed on the international scale.

Emmy Noether’s Zürich lecture can be considered as the high point in her math-ematical career.

371928/29 Emmy Noether had been in Moscow as a visiting professor, on the invitation of Alexandrov whomshe knew from Göttingen.

38The photo is also contained in the Oberwolfach photo collection online. Perhaps the photo was taken byNatascha Artin, the wife of Emil Artin.


4.6 Göttingen exodus: 1933

The year 1933 brought about the almost complete destruction of the unique math-ematical scene in Göttingen. In consequence of the antisemitic political line of theNazi government many scientists of Jewish origin had to leave the university, as wellas those who were known to be critical towards the new government. The Göttingensituation in 1933 has often been described, and so we can refer to the literature, e.g.,[Sch87], [Seg03].

Emmy Noether was of Jewish origin and so she too was a victim of the newgovernment policy. On May 5, 1933 Emmy Noether obtained the message that shewas put “temporarily on leave” from lecturing at the university. When Hasse heardthis, he wrote a letter to her; we do not know the text of his letter but from her replywe may conclude that he asked whether he could be of help. Emmy Noether repliedon May 10, 1933:

Lieber Herr Hasse! Vielen herzlichen Dank für Ihren guten freund-schaftlichen Brief! Die Sache ist aber doch für mich sehr viel wenigerschlimm als für sehr viele andere: rein äußerlich habe ich ein kleinesVermögen (ich hatte ja nie Pensionsberechtigung), sodaß ich erst ein-mal in Ruhe abwarten kann; im Augenblick, bis zur definitiven Entschei-dung oder etwas länger, geht auch das Gehalt noch weiter. Dann wirdwohl jetzt auch einiges von der Fakultät versucht, die Beurlaubung nichtdefinitiv zu machen; der Erfolg ist natürlich im Moment recht fraglich.Schließlich sagte Weyl mir, daß er schon vor ein paar Wochen, woalles noch schwebte, nach Princeton geschrieben habe wo er immernoch Beziehungen hat. Die haben zwar wegen der Dollarkrise jetzt auchkeine Entschlußkraft; aber Weyl meinte doch daß mit der Zeit sich et-was ergeben könne, zumal Veblen im vorigen Jahr viel daran lag, michmit Flexner, dem Organisator des neuen Instituts, bekannt zu machen.Vielleicht kommt einmal eine sich eventuell wiederholende Gastvor-lesung heraus, und im übrigen wieder Deutschland, das wäre mir natür-lich das liebste. Und vielleicht kann ich Ihnen sogar auch einmal so einJahr Flexner-Institut verschaffen – das ist zwar Zukunftsphantasie – wirsprachen doch im Winter davon …

Dear Mr. Hasse! Thanks very much for your good, friendly letter! Butfor myself, the situation is much less dire than for many others: in fact Ihave a small fortune (after all I was never entitled to pension) and hencefor the time being I can quietly wait and see. Also, the salary paymentscontinue until the final decision or even somewhat longer. Moreoverthe Faculty tries to avert my suspension to become final; at the moment,however, there is little hope for success. Finally, Weyl told me thatsome weeks ago already when things were still open, he had written to


Princeton where he still has contacts. At the moment, however, becauseof the dollar crisis they don’t have much freedom there for their decisions;but Weyl believes that in the course of time there may arise something,in particular since Veblen last year was eager to introduce me to Flexner,the organizer of the new Institute. Perhaps there will emerge a visitingprofessorship which may be iterated, and in the meantime Germanyagain, this would be the best solution for me, naturally. And maybe Iwill be able to manage for you too a year in the Flexner Institute – butthis is my fantasy for the future – we have talked last winter about this …

The first impression while reading this letter is her complete selflessness, whichis well-known from other reports on her life and which is manifest here again. Shedoes not complain about her own situation but only points out that for other peoplethings may be worse. Reading further, we see that the Faculty in Göttingen tries tokeep her; this shows that she was respected there as a scientist and teacher althoughshe still did not have a tenured position. Hermann Weyl was a full professor andhence a member of the Faculty committee; we can surely assume that he was one ofthe driving forces in trying to save Emmy Noether for a position in Göttingen. Infact, in his memorial speech [Wey35] Weyl said:

It was attempted, of course, to influence the Ministerium and other re-sponsible and irresponsible bodies so that her position might be saved.I suppose there could hardly have been in any other case such a pile ofenthusiastic testimonials filed with the Ministerium as was sent in onher behalf. At that time we really fought; there was still hope left thatthe worst could be warded off …

And finally, in the above Noether letter we read that, independent of these attempts,Weyl had written to Princeton on her behalf. We do not know whom in PrincetonWeyl had adressed. Since Noether mentions in her letter Veblen and Flexner, it seemsprobable that Weyl had written to one or both of them. Abraham Flexner was thespiritual founder and the first director of the newly-founded Institute for AdvancedStudy in Princeton. Oswald Veblen was the first permanent mathematics professor ofthe IAS. Certain indications suggest that Weyl had written to Lefschetz too; see nextsection. Solomon Lefschetz had the position of full professor at Princeton University.

One year earlier, in the late summer of 1932, Weyl had rejected an offer to join theIAS as a permanent member. But now, since the political situation had deteriorated,he inquired whether it was possible to reverse his decision. (It was.) From Noether’sletter we infer that Weyl did not only write on his own behalf but also on Noether’s.This fact alone demonstrates the very high esteem in which he held Noether as amathematician and as a personality.39

39In the course of time, Weyl used his influence in American academic circles to help many other mathemati-cians as well.


But of course, the best solution would be that Noether could stay in Göttingen.This was what Weyl wished to achieve foremost, as we cited above. (It was in vain.)Weyl reports in [Wey35]:

I have a particularly vivid recollection of these months. Emmy Noether,her courage, her frankness, her unconcern about her own fate, her concil-iatory spirit, were, in the middle of all the hatred and meanness, despairand sorrow surrounding us, a moral solace.

That stormy time of struggle in the summer of 1933 in Göttingen drew them closertogether. This is also evident from the words Weyl used two years later in his speechat her funeral:40

You did not believe in evil, indeed it never occurred to you that it couldplay a role in the affairs of man. This was never brought home to me moreclearly than in the last summer we spent together in Göttingen, the stormysummer of 1933. In the midst of the terrible struggle, destruction andupheaval that was going on around us in all factions, in a sea of hate andviolence, of fear and desperation and dejection – you went your own way,pondering the challenges of mathematics with the same industriousnessas before. When you were not allowed to use the institute’s lecture hallsyou gathered your students in your own home. Even those in their brownshirts were welcome; never for a second did you doubt their integrity.Without regard for your own fate, openhearted and without fear, alwaysconciliatory, you went your own way. Many of us believed that anenmity had been unleashed in which there could be no pardon; but youremained untouched by it all.

Parallel to the attempts of the Faculty to keep Noether in Göttingen, Hasse tookthe initiative and collected testimonials41 which would put into evidence that EmmyNoether was a scientist of first rank and hence it would be advantageous for thescientific environment of Göttingen if she did not leave. Hasse collected 14 suchtestimonials. Together they were sent to the Kurator of the university who was toforward them to the Ministerium in Berlin. Recently we have found the text of thosetestimonials which are kept in the Prussian State archives in Berlin; we plan to publishthem separately. The names of the authors are:

H. Bohr, Kopenhagen

Ph. Furtwängler, Wien

G. H. Hardy, Cambridge

40See Section 4.9.2.41The German word is “Gutachten”. I am not sure whether the translation into “testimonial” is adequate. My

dictionary offers also “opinion” or “expertise” or “letter of recommendation”. I have chosen “testimonial” sinceWeyl uses this terminology.


H. Hasse, Marburg

O. Perron, München

T. Rella, Wien

J. A. Schouten, Delft

B. Segre, Bologna

K. Shoda, Osaka

C. Siegel, Frankfurt

A. Speiser, Zürich

T. Takagi, Tokyo

B. L. van der Waerden, Leipzig

H. Weyl, Göttingen

We see that also Hermann Weyl wrote a testimonial. We have included it in theappendix, translated into English; see Section 4.9.1. Note that Weyl compared EmmyNoether to Lise Meitner, the nuclear physicist. In the present situation this comparisonmay have been done since Meitner, also of Jewish origin, was allowed to stay in Berlincontinuing her research with Otto Hahn in their common laboratory. After all, theinitiatives of Hasse and of Weyl were to obtain a similar status for Emmy Noether inGöttingen.

As is well-known, this was in vain. Perhaps those testimonials were never read af-ter the Kurator of Göttingen University wrote to the Ministerium that Emmy Noether’spolitical opinions were based on “Marxism”.42

Let us close this section with some lines from a letter of Weyl to Heinrich Brandtin Halle. The letter is dated December 15, 1933; at that time Weyl and Noetherwere already in the USA. Brandt was known to be quite sceptical towards abstractmethods in mathematics; he did not even like Artin’s beautiful presentation of hisown (Brandt’s) discovery, namely that the ideals and ideal classes of maximal ordersin a simple algebra over a number field form a groupoid under multiplication.43 (Thenotion of “groupoid” is Brandt’s invention.) Weyl’s letter is a reply to one fromBrandt which, however, is not known to us. Apparently Brandt had uttered somewords against Noether’s abstracting method, and Weyl replied explaining his ownviewpoint:44

… So wenig mir persönlich die “abstrakte” Algebra liegt, so schätzeich doch ihre Leistungen und ihre Bedeutung offenbar wesentlich höher

42See [Tol90]. – By the way, there was another such initiative started, namely in favor of Courant who alsohad been “beurlaubt” from Göttingen University. That was signed by 28 scientists including Hermann Weyl andHelmut Hasse. Again this was not successful, although this time the Kurator’s statement was not as negative asin Noether’s case. (We have got this information from Constance Reid’s book on Courant [Rei76].)

43Artin’s paper is [Art28b].44 I would like to thank M. Göbel for sending me copies of this letter from the Brandt archive in Halle. The

letter is published in [Jen86], together with other letters Brandt-Weyl and Brandt-Noether.


ein, als Sie das tun. Es imponiert mir gerade an Emmy Noether, daß ihreProbleme immer konkreter und tiefer geworden sind.

Personally, the “abstract” algebra doesn’t particularly suit me but ap-parently I do estimate its achievements and importance much higherthan you are doing. I am particularly impressed that Emmy Noether’sproblems have become more and more concrete and deep.

Weyl continues as follows. It is not known whether Brandt had written some com-ments on Noether’s Jewish origin and connected this with her abstract way of think-ing, or perhaps Weyl’s letter was triggered by the general situation in Germany andespecially in Göttingen:

Warum soll ihr, der Hebräerin, nicht zustehen, was in den Händen des“Ariers” Dedekind zu großen Ergebnissen geführt hat? Ich überlassees gern Herrn Spengler und Bieberbach, die mathematische Denkweisenach Völkern und Rassen zu zerteilen. Daß Göttingen den Anspruchverloren hat, mathematischer Vorort zu sein, gebe ich Ihnen gerne zu –was ist denn überhaupt von Göttingen übrig geblieben? Ich hoffe undwünsche, daß es eine seiner alten Tradition würdige Fortsetzung durchneue Männer finden möge; aber ich bin froh, daß ich es nicht mehr gegeneinen Strom von Unsinn und Fanatismus zu stützen brauche!

Why should she, as of Hebrew descent, not be entitled to do what hadled to such great results in the hands of Dedekind, the “Arian” ? I leaveit to Mr. Spengler and Mr. Bieberbach to divide the mathematical wayof thinking according to nations and races. I concede that Göttingen haslost its role as a high-ranking mathematical place – what is actually leftof Göttingen? I hope and wish that Göttingen would find a continuationby new men, worthy of its long tradition; but I am glad that I do not haveto support it against a torrent of nonsense and fanaticism.

4.7 Bryn Mawr: 1933–1935

As we have seen in the foregoing section, Weyl had written to Princeton on behalfof Emmy Noether, and this was in March or April 1933 already. Since he was goingto join the Institute for Advanced Study in Princeton, one would assume that he hadrecommended accepting Emmy Noether as a visiting scientist of the Institute. Weknow that some people at the Institute were interested in getting Noether to Princeton,for at the International Zürich Congress Oswald Veblen had been eager to introduceEmmy Noether to the Institute’s director, Abraham Flexner. (See Noether’s letter toHasse, cited in the foregoing section.)


But as it turned out, Emmy Noether did not receive an invitation as a visitor to theInstitute. We do not know the reason for this; perhaps the impending dollar crisis,mentioned in Noether’s letter to Hasse, forced the Institute to reduce its availablefunds. Or, may there have been other reasons as well? On the other hand, from thedocuments which we found in the archive of Bryn Mawr College it can be seen thatthe Institute for Advanced Study contributed a substantial amount towards the salaryof Emmy Noether in Bryn Mawr.

We do not know who was the first to suggest that Bryn Mawr College could bea suitable place for Emmy Noether. Some evidence points to the conclusion that itwas Solomon Lefschetz. In fact, we have found a letter, dated June 12, 1933 already,adressed to the “Emergency Committee inAid of Displaced German Scholars”, wherehe discusses future aspects for Emmy Noether and proposes Bryn Mawr.45 Lefschetzhad visited Göttingen two years ago and so he knew Emmy Noether personally.Lefschetz’ letter is quite remarkable since, firstly, he clearly expresses that EmmyNoether, in his opinion, was a leading figure in contemporary mathematics; secondlywe see that he had taken already practical steps to provide Bryn Mawr with at leastpart of the necessary financial means in order to offer Emmy Noether a stipend. Letus cite the relevant portions of that letter:

Dear Dr. Duggan: I am endeavoring to make connections with somewealthy people in Pittsburgh, one of them a former Bryn Mawr student,with a view of raising a fund to provide a research associateship at BrynMawr for Miss Emmy Noether. As you may know, she is one of the mostdistinguished victims of the Hitler cold pogrom and she is victimizeddoubly; first for racial reasons and second, owing to her sex. It occuredto me that it would be a fine thing to have her attached to Bryn Mawrin a position which would compete with no one and would be createdad hoc; the most distinguished feminine mathematician connected withthe most distinguished feminine university. I have communicated withMrs. Wheeler, the Head of the Department at Bryn Mawr, and she is notonly sympathetic but thoroughly enthusiastic for this plan.

So far as I know, your organization is the only one which is endeavoringto do anything systematic to relieve the situation of the stranded Germanscientists. As I do not think random efforts are advisable, I wish firstof all to inform you of my plan. Moreover, if I were to succeed onlypartially, would it be possible to get any aid from your organization? Iwould greatly appreciate your informing me on this point at your earliestconvenience.

In the preliminary communication with my intended victims I mentionedthe following proposal: to contribute enough annually to provide Miss

45We have found this letter in the archives of the New York Public Library.


Noether with a very modest salary, say $ 2000, and a retiring allowanceof $ 1200.

Yours very sincerely, S. Lefschetz.

Already one month later the committee granted the sum of $ 2000 to Bryn Mawr forEmmy Noether.

There arises the question from whom Lefschetz had got the information, at thatearly moment already, that Emmy Noether had been suspended.46 We are inclinedto believe that it was Hermann Weyl. I do not know whether the correspondence ofLefschetz of those years has been preserved in some archive, and where. Perhaps itwill be possible to find those letters and check.

Emmy Noether arrived in Bryn Mawr in early November 1933. Her first letterfrom Bryn Mawr to Hasse is dated March 6, 1934. She reported, among other things,that since February she gave a lecture once a week at the Institute for Advanced Studyin Princeton. In this lecture she had started with representation modules and groupswith operators. She mentions that Weyl too is lecturing on representation theory, andthat he will switch to continuous groups later. It appears that the Göttingen situation of1926/27 was repeating. And we imagine Hermann Weyl and Emmy Noether walkingafter her lectures around the Campus of Princeton University47 instead of Göttingen’snarrow streets, vividly discussing new aspects of representation theory.

In the book [Rei76] on Courant we read:

Weyl sent happy letters from Princeton. In Fine Hall, where Flexner’sgroup was temporarily housed, German was spoken as much as English.He frequently saw Emmy Noether …

Perhaps in the Courant legacy we can find more about Weyl and Noether in Princeton,but we have not been able yet to check those sources.

Every week Emmy Noether visited the Brauers in Princeton; Richard Brauer wasassistant to Weyl in that year and perhaps sometimes Hermann Weyl also joined theircompany. The name of Hermann Weyl appears several times in her letters to Hassefrom Bryn Mawr. In November 1934 she reports that she had studied Weyl’s recentpublication on Riemann matrices in the Annals of Mathematics.

Emmy Noether died on April 14, 1935. One day later Hermann Weyl cabled toHasse:

hasse mathematical institute gottingen – emmy noether died yesterday– by sudden collapse after successful – operation of tumor48 few daysago – burial wednesday bryn mawr – weyl

46Emmy Noether had been “beurlaubt”, i.e., temporarily suspended from her duties, in May 1933. Observingthat mail from Europe to USA used about 2–3 weeks at that time, we conclude that Lefschetz must have startedworking on his Noether-Bryn Mawr idea immediately after receiving the news about her suspension. Noetherwas finally dismissed from university on September 9, 1933.

47The Institute’s Fuld Hall had not yet been built and the School of Mathematics of the Institute was temporarilyhoused in Fine Hall on the University Campus.


At the burial ceremony on Wednesday Weyl spoke on behalf of her German friendsand colleagues. We have included an English translation of this moving text in theappendix; see Section 4.9.2. One week later he delivered his memorial lecture inthe large auditorium of Bryn Mawr College. That text is published and well known[Wey35].

4.8 The Weyl–Einstein letter to the NYT

On Sunday May 5, 1935 the New York Times published a “Letter to the Editor”,signed by Albert Einstein and headed by the following title:

Professor Einstein Writes in Appreciation of a Fellow-Mathematician.

We have included the text of this letter in our appendix; see Section 4.9.3.Reading this letter one is struck by the almost poetic style which elevates the text

to one of the pearls in the literature on mathematics. The text is often cited, the lastcitation which I found is in the Mitteilungen der DMV 2007, where Jochen Brüningtries to connect mathematics with poetry [Brü07]. But because of this character ofstyle it has been doubted whether the text really was composed by Einstein himself.If not then this would not have been the first and not the last incident where Einsteinhad put his name under a text which was not conceived by himself – provided thatin his opinion the subject was worth-while to support. Since Weyl’s poetic style wasknown it was not considered impossible that the text was composed by HermannWeyl.

Some time ago I have come across a letter signed by Dr. Ruth Stauffer-McKee. Iinclude a copy of that letter in the appendix; see Section 4.9.4. In particular I referto the last paragraph of the letter. Based on the information provided by Stauffer Icame to the conclusion that, indeed, the text was essentially written by Weyl. I haveexpressed this opinion in my talk in Bielefeld and also in a “Letter to the Editor” ofthe Mitteilungen der Deutschen Mathematiker-Vereinigung [Roq07a].

However, recently I have been informed that Einstein’s draft of this letter in hisown handwriting has been found by Siegmund-Schultze49 in the Einstein archive inJerusalem. The article appeared in the Mitteilungen der DMV, see [SS07]. This thensettles the question of authorship in favor of Einstein. But what had induced RuthStauffer to claim that Weyl had “inspired” Einstein’s letter?

In order to understand Stauffer’s letter let us explain its background.

48President Park of Bryn Mawr had sent a detailed report, dated May 16, 1935, to Otto Nöther in Mannheim,a cousin of Emmy Noether. A copy of that letter is preserved. There it is stated that according to the medicaldiagnosis of the doctors who operated her, Emmy Noether suffered from a “pelvis tumor”.

49I would like to thank R. Siegmund-Schultze for a number of interesting comments and corrections to thisarticle.


In 1972 there appeared a paper on Emmy Noether in the American Mathema-tical Monthly, authored by Clark Kimberling [Kim72]. Among other information thepaper contains the text of Einstein’s letter to the New York Times. Kimberling hadobtained the text from an article in the Bryn Mawr Alumnae Bulletin where it hadbeen reprinted in 1935. Together with that text, we find in [Kim72] the following:

A note in the files of the Bryn MawrAlumnae Bulletin reads, “The abovewas inspired, if not written, by Dr. Hermann Weyl, eminent Germanmathematician. Mr. Einstein had never met Miss Noether.”

(Here, by “above” was meant the text of the Einstein letter to the New York Times.)While the first sentence of that “note” can be considered as an affirmation of the

guess that Weyl had conceived the text of Einstein’s letter, the second sentence ishard to believe. Emmy Noether often visited the Institute for Advanced Study inPrinceton, the same place where Einstein was, and it seems improbable that they didnot meet there. After all, Einstein was already in May 1918 well informed aboutNoether’s achievements, when he wrote to Hilbert praising her work [Noe18]. Andin December that year, after receiving the printed version of this work, he wroteto Felix Klein and recommended her Habilitation. In the 1920s, Einstein had acorrespondence with Emmy Noether who acted as referee for papers which weresubmitted to the Mathematische Annalen. It is hard to believe that in Princeton hewould have avoided meeting Emmy Noether, whom he esteemed so highly. Moreover,we have already mentioned in Section 4.4 that Einstein probably had met Noether in1915 in Göttingen. Also, on the DMV-meetings 1909 in Salzburg and 1913 in Wienboth Einstein and Emmy Noether presented talks and there was ample opportunityfor them to meet.

Thus it seemed that the “note” which Kimberling mentioned had been written bysomeone who was not well informed about the situation in the early thirties. Actually,that “note” was not printed in the Bryn Mawr Alumnae Bulletin but it was added laterby typewriter, maybe only on the copy which was sent to Kimberling. It is not knownwho had been the author of that “note”.

In the same volume of the American Mathematical Monthly where his article[Kim72] had appeared, Kimberling published an Addendum saying that Einstein’sformer secretary, Miss Dukas, had objected to the statement that the letter written byEinstein was “inspired, if not written by Dr. Hermann Weyl”. She insisted that theletter was written by Einstein himself at the request of Weyl.

This, however, induced Ruth Stauffer to write the above mentioned letter to theeditor of the American Mathematical Monthly, which we are citing in Section 4.9.4.Ruth Stauffer had been a Ph.D. student of Emmy Noether in Bryn Mawr and in herletter she recalls vividly the mathematical atmosphere in Princeton at that time.

On this evidence I was led to believe that the statement of Einstein’s secretaryDukas may be due to a mix-up on her part. For, only shortly before Noether’s deathEinstein had written another letter in which he recommends that Emmy Noether’s


situation in Bryn Mawr College should be improved and put on a more solid base.At that time President Park of Bryn Mawr had tried to obtain testimonies on EmmyNoether, which could be used in order to get funds for a more permanent position.50

Einstein’s testimony is dated January 8, 1935 and is written in German; we havefound it in the archives of the Institute for Advanced Study in Princeton. Its full textreads:

Fräulein Dr. Emmy Noether besitzt unzweifelhaft erhebliches schöpfer-isches Talent, was jeweilen von nicht sehr vielen Mathematikern einerGeneration gesagt werden kann. Ihr die Fortsetzung der wissenschaft-lichen Arbeit zu ermöglichen, bedeutet nach meiner Ansicht die Er-füllung einer Ehrenpflicht und wirkliche Förderung wissenschaftlicherForschung.

Without doubt Miss Dr. Emmy Noether commands significant and cre-ative talent; this cannot be said of many mathematicians of one genera-tion. In my opinion it is an obligation of honor to provide her with themeans to continue her scientific work, and indeed this will be a propersupport of scientific research.

It is apparent that the style of this is quite different from the style of the letter tothe New York Times.

Although we now know that Miss Dukas was right and Einstein had composed hisNYT-letter with his own hand, there remains the question as to the basis of Stauffer’scontentions.

Stauffer was a young student and what she reports is partly based on what sheheard from Mrs. Wheeler. But the latter, who was head of the mathematics departmentof Bryn Mawr College at the time, had studied in Göttingen with Hilbert in the sameyears as Hermann Weyl had; so they were old acquaintances and it seems probablethat Weyl himself had told her the story as it had happened. Thus it may well havebeen that first Weyl had sent his obituary on Emmy Noether to the New York Times,and that this was returned with the suggestion that Einstein should write an obituary –as Ruth Stauffer narrates. And then Einstein wrote his letter “at the request of Weyl”,as Miss Dukas has claimed. Whether there was any cooperation between Einsteinand Weyl while drafting the letter is not known. But we can safely assume thatboth had talked if not about the text of the letter but certainly about Emmy Noether’spersonality, her work and her influence on mathematics at large. In this way Stauffer’sclaim may be justified that Weyl had “inspired” Einstein in writing his letter.

Remark. It has been pointed out to me by several people that the very last sentencein the English version of Einstein’s letter deviates in its meaning from the original

50This was successful, but Emmy Noether died before she got to know about it. – Other testimonials, bySolomon Lefschetz, Norbert Wiener and George D. Birkhoff are published in Kimberling’s article [Kim81].


German text wheras otherwise the translation seems to be excellent. 51 In the Englishversion it is said that Noether’s last years in Bryn Mawr were made the “happiest andperhaps most fruitful years of her entire career”, but the German text does not referto her entire career and only pointed out that death came to her “mitten in froher undfruchtbarer Arbeit”. I do not know who had translated the German text into English.There is a letter of Abraham Flexner, the director of the Institute for Advanced Studyin Princeton, addressed to Einstein and dated April 30, 1935, in which Flexner thanksEinstein for the “beautiful tribute to Miss Noether” and continues: “I shall translateit into English and send it to the New York Times, through which it will reach, Ithink, many of those who should know of her career.” But it does not seem justified,I believe, to conclude that Flexner personally did the translation job. He was quitebusy with all kinds of responsibilities and certainly he had contacts to experts whowould have been willing and competent to do it.52

Final remark. Weyl’s solidarity with Emmy Noether extended to her brother andfamily. Emmy’s brother Fritz had emigrated to Russia where he got a position at theuniversity in Tomsk. In 1937 he was arrested and sentenced to 25 years in prisonbecause of alleged espionage for Germany. In the Einstein archive in Jerusalem wehave found a letter, dated April 1938 and signed by Einstein, addressed to the Russianminister of foreign affairs Litvinov. In this letter Einstein appeals to the minister infavor of Fritz Noether, whom he (Einstein) is sure to be innocent. In the Einsteinarchive, right after this letter, is preserved a curriculum vitae of Fritz Noether inWeyl’s handwriting. Thus again it appears that Weyl has “inspired” Einstein to writesuch a letter.53

Among Weyl’s papers I found a number of letters from 1938 and the followingyears, which show that he cared for the two sons of Fritz Noether, Hermann andGottfried, who had to leave the Soviet Union after their father had been sentenced.Weyl saw to it that they obtained immigrant visa to the United States, and that theygot sufficient means to finance their university education. Both became respectedmembers of the scientific community.

4.9 Appendix: documents

4.9.1 Weyl’s testimony

The following text54 is from the testimonial, signed by HermannWeyl on July 12, 1933and sent by Hasse to the Ministerium in Berlin together with 13 other testimonials.

51The German text is published in my “Letter to the Editor” [Roq07a].52Siegmund-Schultze [SS07] advocates reasons to assume that indeed, Flexner himself did the translation job.53The appeal of Einstein was in vain. In 1941, when German troops were approaching the town of Orjol where

Fritz was kept in prison, he was sentenced to death and immediately executed. See, e.g., [Sch91].54Translated from German by Ian Beaumont.


We have found these testimonials in the Prussian state archive Berlin.

Emmy Noether has attained a prominent position in current mathematical research– by virtue of her unusual deep-rooted prolific power, and of the central importance ofthe problems she is working on together with their interrelationships. Her research andthe promising nature of the material she teaches enabled her in Göttingen to attract thelargest group of students. When I compare her with the two woman mathematicianswhose names have gone down in history, Sophie Germain and Sonja Kowalewska, shetowers over them due to the originality and intensity of her scientific achievements.The name Emmy Noether is as important and respected in the field of mathematicsas Lise Meitner is in physics.

She represents above all “AbstractAlgebra”. The word “abstract” in this context inno way implies that this branch of mathematics is of no practical use. The prevailingtendency is to solve problems using suitable visualizations, i.e. appropriate formationof concepts, rather than blind calculations. Fräulein Noether is in this respect thelegitimate successor of the great German number theorist R. Dedekind. In addition,Quantum Theory has made Abstract Algebra the area of mathematics most closelyrelated to physics.

In this field, in which mathematics is currently experiencing its most activeprogress, Emmy Noether is the recognised leader, both nationally and internation-ally.

Hermann Weyl

4.9.2 Weyl’s funeral speech

Hermann Weyl had read his funeral speech on April 18, 1935 at the funeral ceremonyin the house of President Park in Bryn Mawr. For the text see Chapter 3, Section 3.2,page 123.

4.9.3 Letter to the New York Times

The following text was published on Sunday, May 5, 1935 by the NewYork Times, withtheheading: “ProfessorEinsteinWrites inAppreciationof aFellow-Mathematician”.

To the Editor of The New York Times:

The efforts of most human-beings are consumed in the struggle for their dailybread, but most of those who are, either through fortune or some special gift, relievedof this struggle are largely absorbed in further improving their worldly lot. Beneaththe effort directed toward the accumulation of worldly goods lies all too frequentlythe illusion that this is the most substantial and desirable end to be achieved; but thereis, fortunately, a minority composed of those who recognize early in their lives that


the most beautiful and satisfying experiences open to humankind are not derived fromthe outside, but are bound up with the development of the individual’s own feeling,thinking and acting. The genuine artists, investigators and thinkers have always beenpersons of this kind. However inconspicuously the life of these individuals runs itscourse, none the less the fruits of their endeavors are the most valuable contributionswhich one generation can make to its successors.

Within the past few days a distinguished mathematician, Professor Emmy Noether,formerly connected with the University of Göttingen and for the past two years at BrynMawr College, died in her fifty-third year. In the judgment of the most competent liv-ing mathematicians, Fräulein Noether was the most significant creative mathematicalgenius thus far produced since the higher education of women began. In the realm ofalgebra, in which the most gifted mathematicians have been busy for centuries, shediscovered methods which have proved of enormous importance in the developmentof the present-day younger generation of mathematicians. Pure mathematics is, inits way, the poetry of logical ideas. One seeks the most general ideas of operationwhich will bring together in simple, logical and unified form the largest possiblecircle of formal relationships. In this effort toward logical beauty spiritual formulasare discovered necessary for the deeper penetration into the laws of nature.

Born in a Jewish family distinguished for the love of learning, Emmy Noether,who, in spite of the efforts of the great Göttingen mathematician, Hilbert, neverreached the academic standing due her in her own country, none the less surroundedherself with a group of students and investigators at Göttingen, who have alreadybecome distinguished as teachers and investigators. Her unselfish, significant workover a period of many years was rewarded by the new rulers of Germany with adismissal, which cost her the means of maintaining her simple life and the opportunityto carry on her mathematical studies. Farsighted friends of science in this country werefortunately able to make such arrangements at Bryn Mawr College and at Princetonthat she found inAmerica up to the day of her death not only colleagues who esteemedher friendship but grateful pupils whose enthusiasm made her last years the happiestand perhaps the most fruitful of her entire career.

Albert Einstein.Princeton University, May 1, 1935.

4.9.4 Letter of Dr. Stauffer-McKee

The following letter was sent by Dr. Ruth Stauffer-McKee on October 17, 1972 to theeditor of the American Mathematical Monthly, Professor H. Flanders. A carbon copyhad been sent to Professor Kimberling. I am indebted to Clark Kimberling for givingme access to his private archive.55

55The Kimberling archive on Emmy Noether is now contained in the Handschriftenabteilung of GöttingenUniversity.


Dear Mr. Flanders,

After reading the Addendum to “Emmy Noether” in the August September issueof the American Mathematical Monthly, I was much disturbed by the apparent lackof information concerning the thirties at Princeton! Rechecking the reference to theoriginal article which appeared in February 1972 I was even more disturbed to notethat the quote was attributed to a note in the files of Bryn Mawr Alumnae Bulletin.A telephone conversation and a careful check by the Staff of the Bulletin assured methat there was nothing in the files of the Bulletin to even imply that “Mr. Einstein hadnever met Miss Noether.”

In respect to the “thirties at Princeton”, I should like to note that there was an airof continued excitement at the Institute for Advanced Study. Solomon Lefschetz, aguiding spirit who worked diligently to help the displayed mathematicians, HermannWeyl, a leading mathematician of that time who had learned to know Miss Noetherin Göttingen, and John von Neumann, then considered a brilliant young genius, wereall at the Institute when Einstein arrived in December of 1933. Mrs. Wheeler, ofBryn Mawr, often told of the welcoming party which she and Miss Noether attended.

Mrs. Wheeler usually drove Miss Noether to Princeton for lectures and includedMiss Noether’s students in the parties. We listened to talks by these men who were theleaders in new exciting theories. It was a friendly group and after the talks everyonegathered for more talk and coffee in a long pleasant common room. There is no doubtthat Einstein and Noether were acquainted. I saw them in the same group!

As regards the quote in the “addendum to ‘Emmy Noether’” “inspired, if notwritten by Dr. Hermann Weyl” is certainly true. The writing of the obituary was avery natural occurrence. Hermann Weyl was considered by the mathematicians as themathematical leader of the time and at the peak of his productivity and he had probablythe greatest knowledge and understanding of her work. Einstein had begun to slowdown and von Neumann was relatively young and still growing. It was, therefore,obvious to all the mathematicians that Weyl should write the obituary – which he did.He, furthermore, sent it to the New York Times, the New York Times asked who isWeyl? Have Einstein write something, he is the mathematician recognized by theworld. This is how Einstein’s article appeared. It was most certainly “inspired” byWeyl’s draft. These facts were told to me at the time by Mrs. Wheeler who wasindignant that the New York Times had not recognized the mathematical stature ofHermann Weyl.

Very truly yours,Ruth Stauffer McKee

Senior Mathematician

Chapter 5

Emmy Noether: The testimonials

Translation of the article:

Emmy Noether: Die Gutachten. Einige neue Noetheriana

Mitteilungen der Deutschen Mathematiker Vereinigung, Vol. 16 (2008), 118–123.

5.1 Preface 163

5.2 The testimonials 165

5.3 The accompanying letters 168

5.4 The petition of students 170

5.5 The American testimonials 172

5.1 Preface

In the course of our work to edit the Hasse–Noether correspondence1 we have foundsome documents which perhaps deserve independent interest for the history of math-ematics and of mathematicians. Some of these “Noetheriana” have been alreadypublished.2 The present article concerns the testimonials for Emmy Noether from theyears 1933/34. Recently I have discovered the originals of these testimonials.

Remark. I have chosen the English word “testimonial” as the translation of the Ger-man “Gutachten” but I am not quite sure whether this translation reflects the meaningof the German in an unambiguous manner. In any case, the “testimonials” which arediscussed here are statements of mathematicians, written in the years 1933/34, assess-ing the importance of Emmy Noether’s work and the impact of her ideas. Perhaps theword “opinion” or “assessment” would be a little more to the point. I have decided infavor of “testimonial” since Hermann Weyl had used this expression in his MemorialAddress for Emmy Noether which he delivered in Bryn Mawr on April 26, 1935.

1This has appeared in the meantime: [LR06].2See [Roq07b, Roq08]. (Chapters 3 and 4.)

164 5 Emmy Noether: The testimonials

There he said: “I suppose there could hardly have been in any other case such a pileof enthusiastic testimonials filed with the Ministerium as was sent in her behalf.”3

The existence of those testimonials was known not only from Weyl’s memorialaddress but also from the letters of Emmy Noether to Hasse in the summer of 1933.4

Let us recall: On January 30, 1933 Hitler became Reichskanzler of Germany andalready on April 7 the so-called Beamtengesetz was decreed which, among otherconsequences, prohibited people of Jewish descent or those with oppositional politicalviews to work at state universities. Accordingly Emmy Noether was “temporarily”dismissed. She had to complete, within two weeks, a questionnaire inquiring abouther “Aryan” descent as well as her membership in political parties in previous years.There was the danger that she would be permanently dismissed. But some people stillhoped that it would be possible to obtain an exceptional permit which would allowher to stay in Göttingen and continue her work.

In this situation Helmut Hasse, at that time in Marburg, tried to solicit testimonialsfrom prominent mathematicians all over the world, to show that Emmy Noether wasone of the leading mathematicians and hence it would be a great loss for mathematicsin Göttingen and in Germany if she were forced to emigrate.

Until recently the text of those testimonials was unkown, and not even the namesof their senders were known. Günther Frei reports in [Fre77] that Hasse had collectedsuch testimonials and sent them to the Ministerium. In the report of Schappacher[Sch87] on the Göttingen Mathematical Institute it is only said that there were 14 suchtestimonials. We read the same in the extensive and informative writings of CordulaTollmien on Emmy Noether [Tol90]. At first my own search for these testimonialswas also unsuccessful. The archives of Göttingen University did not contain moreinformation about this issue.

But in June 2006 I visited (for another project) the “Geheimes StaatsarchivPreußischer Kulturbesitz” in Berlin. On this occasion I inquired again about the pos-sible whereabouts of the personnel records on Emmy Noether. It turned out that justrecently some material had been received including a dossier on Emmy Noether.5

Apparently these documents had been moved during the war from Berlin to someother place and only now had it been possible to integrate it again into that archive.

This file contained, among other documents, the missing 14 testimonials. Moreprecisely there are only 13 testimonials since one of them is signed jointly by twomathematicians, Harald Bohr and G. H. Hardy. I do not know whether Hasse hadwritten to even more mathematicians who had not answered or at least not answeredin time. In any case I find the list of names of mathematicians remarkable. It testifiesto the high esteem which Emmy Noether met within the mathematical communityof the time. I believe this will be of general interest, which is why I am writing thisarticle.

3Weyl’s text has been reprinted in Auguste Dick’s biography of Emmy Noether [Dic70].4See [LR06], in particular the letters between May 10 and Sep 13, 1933.5GStA PK, I. HA Rep. 76 Kultusministerium, Nr. 10081.


Emmy Noether 1933 at the train station in Göttingen on her departure

However, because of lacking space I cannot show here the full text of those testi-monial letters. Copies of the originals (mostly in German) can be seen and downloadedfrom my homepage: www.roquette.uni-hd.de.

5.2 The testimonials

The testimonials were written as letters addressed to Hasse but obviously they wereformulated in such a way that they could be presented to the official people in thegovernment. Here is the list of mathematicians who had sent testimonials:

1. H. Bohr (Kopenhagen) and G. H. Hardy (Cambridge):2. Ph. Furtwängler (Wien)3. H. Hasse (Marburg)4. O. Perron (München)5. T. Rella (Wien)

http://www.roquette.uni-hd.de


6. J. A. Schouten (Delft)7. B. Segre (Bologna)8. K. Shoda (Osaka)9. C. L. Siegel (Frankfurt)

10. A. Speiser (Zürich)11. T. Takagi (Tokio)12. B. L. van der Waerden (Leipzig)13. H. Weyl (Göttingen)

In all these testimonials it is stressed that, as Perron, for instance, writes:

Emmy Noether gehört zu den führenden Persönlichkeiten in der moder-nen Mathematik …

Emmy Noether belongs to the leading figures in contemporary mathe-matics …

It is due to her and her school that (after Bohr):

die Algebra eine neue Blüte erlebt hat und in der ganzen mathematischenWelt an führender Stelle steht und ihren Bereich weit ausdehnen konntein geometrische und andere Forschungsgebiete hinein …

Algebra has seen a new blossoming and ranks in the mathematical worldas a leading force; it has widely expanded its domain into geometric andother fields of research …

Again and again her impact upon the younger generation of mathematicians is stressed;in this respect Weyl writes:

Sie wusste in Göttingen durch ihre Forschung und durch die Suggesti-vität ihrer Lehre den grössten Kreis von Schülern um sich zu versam-meln …

She was able to collect in Göttingen the largest circle of students, throughher research and the persuavive power of her teaching …

and Furtwängler:

Sie hat auch durch ihre selbstlose und nur von idealen Zielen geleit-ete Lehrtätigkeit einen grossen Kreis von Schülern herangebildet, diesich heute bereits einen geachteten Namen in der mathematischen Weltgemacht haben …

Through her generous teaching which was exclusively motivated byidealistic goals, she has educated a large circle of students who havealready achieved a respected standing in the mathematical world …


After closer examination of the testimonials it strikes me that the illegitimacy ofthe removal of Jewish scientists from German universities is never mentioned, neitherfrom the judicial and political point of view nor under scientific and humanitarianaspects.An exception is the testimonial of Schouten who indeed expresses this clearly:

Es wäre ein grosser Skandal wenn eine solche Kraft wegen Rassen-vorurteil abgebaut würde. Man macht sich in Deutschland anscheinendkeine Vorstellung davon wie empört das deutschfreundliche Auslandüber solche Sachen ist …

It would be a great scandal if such a person is removed due to racialprejudice. Apparently in Germany one has no idea how outraged peo-ple in foreign countries are about such things – people who otherwiseconsider themselves to be pro-German …

The same point is taken up by Takagi:

Es wäre schade, wenn … ihr die venia legendi an der Universität Göt-tingen beraubt wird, zumal wegen eines Umstandes, woran sie keineSchuld trägt !

It would be a shame if … she would be deprived of her venia legendi atthe University of Göttingen, in particular if this were because of a factwhich she cannot be blamed for !

Otherwise, while reading the testimonials one may get the impression that thesituation was not particularly upsetting. In essence these testimonials could have beenwritten in the same way if, for instance, Noether would have received a tempting offerfrom a university in a foreign country and therefore one should try to make her stayin Göttingen. Perhaps the referees had underestimated the seriousness of the politicalsituation. Or, maybe this academic reservation can be explained by the fact that onewished to achieve something and therefore tried to avoid affronting the ruling politicalforces. Certainly, this last motive is the reason why in some testimonials it is stressedthat Emmy Noether’s work is important for mathematical science “in Germany”.Clearly, all referees agreed that mathematics is international and cannot be reducedto national borders. As an example we cite van der Waerden:

Aus aller Welt kamen vor ihrer Beurlaubung die Algebraiker nach Göt-tingen um ihre Methoden zu lernen, ihren Rat zu holen, unter ihrerFührung zu arbeiten …

Before Emmy Noether had been dismissed, algebraists from all over theworld came to Göttingen in order to learn her methods, to get counselfrom her and to work under her guidance …

and Carl Ludwig Siegel:


Insbesondere hat die sog. Theorie der hyperkomplexen Systeme durchVeröffentlichungen und Vorlesungen von Frl. Noether so grosse Förde-rung erfahren, dass die daran anschliessenden Probleme jetzt bei denAlgebraikern der ganzen Welt im Vordergrund des Interesses stehen …

In particular the so-called theory of hypercomplex systems has greatlyadvanced through the publications and lectures of Miss Noether. Thesubsequent problems are now of primary interest among algebraistsaround the world …

This statement by Siegel is remarkable indeed since he had never hidden his disgustof the abstract methods which Noether had advocated. He regarded them as a sign ofdeterioration of mathematics. But he seems to exempt the theory of “hypercomplexsystems”, i.e., algebras. Already on December 9, 1931 he had sent to Hasse hiscompliments for the theorem on cyclic algebras over number fields which had beenjointly discovered by Hasse, Richard Brauer and Emmy Noether. At that time he hadwritten: “The pessimistic outlook which I generally have towards mathematics hasbeen shaken once again …”. Siegel

5.3 The accompanying letters

Besides the testimonials I found also the accompanying letters from Hasse to thecurator of Göttingen University, who was named Valentiner. (His job in this case wasto forward the testimonials to the Ministerium and to add his own opinion.) These arethe following documents:6

1. Hasse to curator on June 3, 1933. Hasse announced that after consultingProfessor Neugebauer (Göttingen) he had started to solicit testimonials forEmmy Noether. Apparently this announcement was sent in order to forestallfinal decisions about the dismissal of Emmy Noether before the testimonialshad been received. Hasse expressed his hope that Emmy Noether could remainin Germany

durch eine Lehrbeauftragung mit Spezialvorlesungen …through a teaching appointment for specialized courses …

Indeed, this covered precisely her past activities in Göttingen. Emmy Noetherhad approved this wording in a letter to Hasse of June 26, 1933.

2. Hasse to curator on July 31, 1933. This letter accompanied the testimonialswhich Hasse had sent the same day. Hasse emphasizes that the teaching ofEmmy Noether would be directed only

6These letters are already cited by Tollmien since copies had also been found in the archive of GöttingenUniversity.


to a relatively small group of students most of them aspiring to anacademic position.

Perhaps this was said to point out that her presence would not come intoconflict with the student mass organizations which were indoctrinated by theNazi propaganda.

3. Curator to Ministry onAugust 7, 1933. The curator states that he is informedabout the scientific standing of Ms. Emmy Noether but adds:

To my knowledge, politically Ms. Noether had stood on Marxistground since the revolution of 1918 until today … regardless of myhigh esteem of the scientific standing of Ms. Noether I feel unableto advocate her case.

According to Tollmien the allegation of “Marxist” ideology covers, followingthe terminology used at the time, all leftist non-communist parties includingthe Social Democrats.

Actually, neither the endorsing testimonials of the mathematicians nor the criticalstatement of the curator had any visible effect. The whole action of dismissing sci-entists of Jewish descent, or of politically critical opinion, was done for ideologicaland political reasons. The people in power who were responsible for this action hadforeseen and clearly accepted the negative consequences for science, as well as thedecline of reputation of Germany in the international world – and they were deter-mined to complete their action ruthlessly. However, some people who were directly orindirectly affected seem to have believed that reason would prevail in due course andobjective arguments would again be taken into account. We know that, for instanceHasse wrote to his friend Davenport on May 1933:

… hope that reason will come back in due course.

and Emmy Noether to Hasse also in May 1933:

… wird ja wohl ziemlich bald eine Beruhigung kommen!

… probably there will pretty soon come a time of slow-down!

As a side remark let us mention that Hasse had also written to Tornier, asking toadvocate Noether’s case at the government agencies. Hasse knew that Tornier hadouted himself as a member of the Nazi party after the Nazis had come to power, andthat Tornier had gained a certain influence on people in power. In any case there existsa letter of Tornier to Hasse, dated September 23, 1933, in which he warns Hasse “forGod’s sake”7 not to send the testimonials to the Ministerium, for:

7“um Gottes willen”


Durch solche Methoden erreichen Sie im günstigsten Falle, dass Sie inallen realen Dingen der Regierung gegenüber absolut einflusslos wer-den, im ungünstigsten Falle können die Folgen für Sie selbst sehr, sehrschlimm sein.

With such methods you can at best achieve that you will have no influencewhatever on the government, but in the worst case the consequences foryourself could be very, very severe.

I do not know whether and how Hasse has reacted to this threat. Anyhow, thetestimonials had already been sent to Berlin in August, and Emmy Noether hadalready lost her venia legendi 8 on September 13, 1933. Nevertheless Emmy Noetherand apparently Hasse too had some hope. This may have been the reason why Hassehad approached Tornier. Hermann Weyl reports in his obituary on Emmy Noether:

It was attempted, of course, to influence the Ministerium and other re-sponsible and irresponsible but powerful bodies so that her positionmight be saved. At that time we really fought …

It appears that Tornier belonged to the “irresponsible bodies” which Weyl mentioned.In his letter Tornier had posed as a kind of a high-ranking officer in the Nazi hierarchy.9

5.4 The petition of students

There was also a petition of 12 students supporting Emmy Noether, but of course thishad no effect either. We know of this petition from Emmy Noether’s letter to Hasseof June 26, 1933. There she writes:

Wichmann hat dem Kurator noch gerade, als dieser Pfingsten nach Berlinfuhr, die Studentenunterschriften – es waren wesentlich die Algebraiker– gegeben …

Wichmann has given the student signatures to the curator, just before thelatter went to Berlin – they were essentially from the algebraists …

Wichmann had been one of her last Ph.D. students.10 The text of this petitionhas been published by Tollmien already, but the names of the undersigned were not

8The permission to deliver lectures at the university.9Reichsobmann fürMathematik imFührerrat derReichsfachschaft N-S-Hochschullehrer undWissenschaftler.

10The thesis of Wichmann was published 1936 in the Monatshefte für Mathematik und Physik. It contains,among other results, a simple proof of the functional equation of Hey’s zeta function of a simple algebra, by meansof reduction to the functional equation of the center (which is assumed to be a number field). It is remarkablethat in this publication, in the year 1936, Wichmann dared to thank Emmy Noether for her counsel. – But in thefunctional equation there remained an undetermined sign which Wichmann could not handle. Recently FalkoLorenz has discussed this beautiful but almost forgotten result and presented it in a lucid manner. At the sametime Lorenz showed that the sign in question can be determined using Hasse’s sum formula for the invariants ofthe algebra [Lor08b].


known yet. The original which I found carries the signatures. All but one of the namescould be identified. These names are as follows:

1. E. Bannow2. E. Knauf3. Tsen4. W. Vorbeck5. G. Dechamps6. W. Wichmann7. H. Davenport (Cambridge, Engl.)8. H. Ulm9. L. Schwarz

10. Walter Brandt (?)11. D. Derry12. Wei-Liang Chow

Biographical information on these mathematicians can be found in the standardliterature (some of them online) and we do not have to present them here.11 Informa-tion concerning the unidentified no. 10 is welcomed.

Tollmien has already observed that in the petition it was attempted to help EmmyNoether by pointing out that the undersigned students are of “Aryan” descent. In viewof the above mentioned names of the undersigned this appears as a tactical move sinceone wanted to achieve something. It is apparent that the undersigned did not agreewith the division of mathematics into an “Aryan” and a “non-Aryan” part. But whenthey say that Emmy Noether

niemals politischen Einfluss auf ihre Schüler ausgeübt hat,

has never exerted political influence on her students,

then this cannot be regarded as a tactical move, considering all that is known abouther. Here I agree with Tollmien.

By the way: For the summer term 1933 Emmy Noether had announced a lecturewith the title: “Hypercomplex methods in number theory”. Most probably this meantthe proof of Artin’s Reciprocity Law which Hasse had given a year earlier by meansof the theory of algebras and dedicated it to her on her 50th birthday. (Hasse’s paperhad just appeared, in spring 1933.) This lecture could not be given anymore in thesummer term 1933 but Emmy Noether then invited the interested students into herapartment to discuss the proof there. The above list of signatures shows that she hadstudents of high level. They wrote that her aim was not to teach single theorems andresults but “vision and understanding” of the theory. And they continue that

11In this connection see also the list of doctoral students in mathematics 1907–1945 at German universities,prepared by Renate Tobies [Tob06].


ihre Vorlesungen alle ihre Schüler mit Begeisterung und Leidenschaftfür die Mathematik erfüllt haben …

her lectures have filled all her students with enthusiasm and excitementfor mathematics …

This text, I admit, has impressed me very much. This is unfeigned. We can safelyassume that also her students from previous years had felt this way. For any teacher,can there a more convincing praise by one’s students? All the testimonials on Noetherhave the same assertion but here we have the statement directly from her students.

In this connection we would like to mention the text of the commemoration speechby Professor Marguerite Lehr which had been delivered on April 18, 1935, four daysafter Emmy Noether’s death, at the Chapel of Bryn Mawr. This speech contained adescription of the lively contact of Emmy Noether with the students of the college.12

5.5 The American testimonials

As said above, the testimonials collected by Hasse are impressive documents for theesteem which Emmy Noether enjoyed among the mathematicians worldwide. Butthe picture would not be complete if the testimonials written in the USA were notmentioned. These were aimed at securing a position for Emmy Noether at Bryn Mawrwhere she would be able to continue her work. I am showing here three testimonialswhich I have found inAmerican archives.Although these testimonials have been citedalready in Kimberling’s article [Kim81] I believe that they are of interest also in thepresent context. The authors of these testimonials are:

1. S. Lefschetz (Princeton)

2. G. D. Birkhoff (Harvard)

3. N. Wiener (M.I.T.)

It is quite remarkable that the importance of Emmy Noether had been clearlyrecognized in the USA at that time already – at least among the leading mathemati-cians. The authors vehemently support her case. Of course, those testimonials werenot meant to impress representatives of a repressive government as it was the case inGermany, and hence one could write quite clearly. But even when considering thisfact the writings seem to me quite remarkable.

Thus Lefschetz writes:13

12The text is published in [Roq07b]. (See Chapter 3.)13Emergency Committee in Aid of Displaced Foreign Scholars records. Manuscripts and Archives Division,

The New York Public Library. Astor, Lenox and Tilden Foundations.


… she is the holder of a front rank seat in every sense of the word.As the leader of the modern algebra school, she developed in recentGermany the only school worthy of note in the sense, not only of isolatedwork, but of very distinguished group scientific work. In fact, it is noexaggeration to say that without exception all the better young Germanmathematicians are her pupils.

Birkhoff comments:14

She is generally regarded as one of the leaders in modern Algebraic The-ory. Within the last ten or fifteen years she and her students in Germanyhave led the way much of the time.

And Wiener:15

Miss Noether is a great personality; the greatest woman mathematicianwho has ever lived; and the greatest woman scientist of any sort nowliving … Leaving all questions of sex aside, she is one of the ten or twelveleading mathematicians of the present generation in the entire world andhas founded what is certain to be the most important close-knit group ofmathematicians in Germany – the Modern School of Algebraists … Ofall the cases of German refugees, whether in this country or elsewhere,that of Miss Noether is without doubt the first to be considered.

These testimonials were written not in the year 1933 but one year later, in 1934/35when it was time to renew the temporary position of Emmy Noether at Bryn Mawr(or even convert it into a permanent position if possible). This was not quite trivialsince “as far as undergraduate work is concerned, she will be probably of no useat Bryn Mawr” (as G. D. Birkhoff observed in one his letters). The excellence ofEmmy Noether rested in research and in high level teaching. Nevertheless, everybodyincluding President Park of Bryn Mawr promoted vehemently the continuation ofEmmy Noether’s appointment there. Eventually it turned out that Emmy Noether’sappointment was secured for some time (not without some financial help from theInstitute for Advanced Study in Princeton) – but unfortunately she did not live to hearabout this.

The efforts to get Emmy Noether to Bryn Mawr had originally started in thesummer of 1933. Reading the correspondence of people and organizations of the timewe are impressed by the quick and unprejudiced way in which help was organized forthe academics who had to emigrate from Germany. In those early letters we also findassessments in various forms of Emmy Noether’s work, but at that time there wereno testimonials in the formal sense since decisions had to be obtained quite quickly.

14Archives of Bryn Mawr College.15Norbert Wiener Papers (MC 22). Institute Archives and Special Collections, MIT Libraries, Cambridge,

Massachusetts.


The first document in support of Emmy Noether which we have found is a letter byS. Lefschetz. (We found it in the NewYork Public Library.) The letter is dated June 12,1933 and is addressed to an organization in New York (c/o Dr. S. P. Duggan) which,Lefschetz says, “is endeavoring something systematic to relieve the situation of thestranded German scientists”. At that time Lefschetz had already talked to ProfessorAnna Wheeler, Head of the Mathematics Department in Bryn Mawr, with the aim ofsecuring a place of work for Emmy Noether, for instance a research associateship.Lefschetz writes that Wheeler “is not only sympathetic but thoroughly enthusiastic”to this plan. In addition, Lefschetz writes, he had contacted some wealthy people inPittsburgh with a view towards raising a fund for that purpose.

It is remarkable how quickly Lefschetz had taken the intitiative in support ofEmmy Noether – at a time when people in Europe like, for instance Hasse and Weyl(and also Emmy herself), still had hopes to be able to keep her in Göttingen. It seemsthat Lefschetz and his colleagues in USA saw the situation more realistically than themathematicians in Germany.

Remark. At this point we would like to draw the reader’s attention to the archiveof Clark Kimberling in which he has collected documents of his research on EmmyNoether. This archive is now kept by the Handschriftenabteilung of the UniversityLibrary at Göttingen and is open for historical research. The mentioned letter ofLefschetz is available there too.

Kapitel 6

Abraham Robinson and his infinitesimals

Revised version of:

Numbers and Models, Standard and Nonstandard.

Mathematische Semesterberichte 57 (2010), 185–199.

The following is a somewhat extended manuscript for a talk at the “Algebra Days”, May 2008,in Antalya. I talked about my personal recollections of Abraham Robinson.

6.1 How I met Abraham Robinson 1756.2 What is nonstandard analysis? 1776.3 Robinson’s visits 1836.4 Nonstandard algebra 1856.5 Nonstandard arithmetic 187

6.1 How I met Abraham Robinson

It was in the early months of 1963. I was visiting the California Institute of Tech-nology on my sabbatical. Somehow during this visit I learned that one year agoWim Luxemburg had given a lecture on A. Robinson’s theory of infinitesimals andinfinitely large numbers. Luxemburg was on leave but I got hold of his Lecture Notes[Lux62]. Although the topic was somewhat distant from my own work I got interestedand, after thorough reading I wished to meet the person who had been able to putLeibniz’ infinitesimals on a solid base and build the modern analysis upon it.

At that time Abraham Robinson was at the nearby University of California at LosAngeles, and I managed to meet him there. I remember an instructive discussionabout his theory which opened my eyes for the wide range of possible applications;he also showed me his motivations and main ideas about it.

Perhaps I am allowed to insert some personal words explaining why I had been soexcited about the new theory of infintesimals. This goes back to my school days inKönigsberg, when I was 16. At that time the school syllabus required that we were to

176 6 Abraham Robinson and his infinitesimals

be instructed in Calculus or, as it was called in German, in Diffentialrechnung. OurMath teacher at that time was an elderly lady who had been retired already but wasreactivated again for school work in order to fill the vacancy of our regular teacher;the latter had been drafted to the army. (It was war time: 1944.) I still remember thesight of her standing in front of the blackboard where she had drawn a wonderfullysmooth parabola, inserting a secant and telling us that y

xis its slope, until finally

she convinced us that the slope of the tangent is dydx

where dx is infinitesimally smalland dy accordingly.

This, I admit, impressed me deeply. Until then our school Math had consistedlargely of Euclidean geometry, with so many problems of constructing triangles fromsome given data. This was o.k. but in the long run that stuff did not strike me asto be more than boring exercises. But now, with those infinitesimals, Math seemedto have more interesting things in stock than I had met so far. And I decided that Iwould study Mathematics if I survived the dangers of war which we knew we wouldbe exposed to very soon. After all, I wanted to find out more about these wonderfullystrange infinitesimals.

Abraham Robinson

Well, I survived. And I managed to enter University and start with Mathematics.The first lecture I attended to was Calculus, with Professor Otto Haupt in Erlangen.There we were told to my disappointment that my Math teacher had not been up todate after all. We were warned to beware of infinitesimals since they do not exist, andin any case they lead to contradictions. Instead, although one writes dy

dxthen this does

not really mean a quotient of two entities, but it should be interpreted as a symbolicnotation only, namely the limit of the quotients y

x.

I survived this disappointment too. Later I learned that dy and dx can be in-terpreted, not as infinitesimals but as some entities of an abstract construction called


differential module, and if that module is one-dimensional then the quotient dydx

wouldmake sense and yield what we had learned anyhow. Certainly, this sounded nice butin fact it was only an abstract frame ignoring the natural idea of infinitesimally smallnumbers.

So when I learned about Robinson’s infinitesimals, my early school day experi-ences came to my mind again and I wondered whether that lady teacher had not beenso wrong after all.

The discussion with Abraham Robinson kindled my interest and I wished to knowmore about it. Some time later there arose the opportunity to invite him to visitus in Germany where he gave lectures on his ideas, first in Tübingen and later inHeidelberg, after I had moved there.

Before continuing with this let me briefly explain what I am talking about, i.e.,Robinson’s theory of nonstandard analysis.

6.2 What is nonstandard analysis?

6.2.1 A preliminary Axiom

Consider the hierarchy of numbers which we present to our students in their first year:

N � Z � Q � R:

Everything starts with the natural numbers N which, due to Kronecker, are “cre-ated by God” (or whatever is considered to be equivalent to Him). The rest is con-structed by mankind, i.e., by the minds of mathematicians. In each step, the structurein question is enlarged such as to admit greater flexibility with respect to some oper-ations defined in the structure. In Z the operation of subtraction is defined such thatZ becomes an additive group; in fact Z is a commutative ring without zero divisors.In Q the operation of division is defined such that Q becomes a field. Finally, in Revery Cauchy sequence is convergent, such that R becomes a complete ordered field.In each step we tell our students that the respective enlargement exists and we explainhow to construct it.

In order to develop what nowadays is called “analysis” the construction usuallystops with the real field R; this is considered to be adequate and quite sufficient as abasis for (real) analysis. But it had not always been considered to be that way. SinceLeibniz had used the natural idea of infinitesimals to build a systematic theory withit, many generations of mathematicians (including my lady teacher) had been taughtin the Leibniz way. Prominent people like Euler, the Bernoullis, Lagrange and evenCauchy (to name only a few) did not hesitate to use them.

Leibniz’ idea was to work in a further enlargement:

R � �R


such that the following Axiom is satisfied. In order to explain the main idea I willfirst state the Axiom in a preliminary form which, however, will not yet be sufficient.Later I will give the final, more general form.

Axiom (preliminary form).

(1) �R is an ordered field extension of R.

(2) �R contains infinitely large elements.

An element ! 2 �R is called “infinitely large” if j!j > n for all n 2 N. Part (2) saysthat the ordering of �R does not satisfy the axiom of Archimedes.

Fields with the properties (1) and (2) were known for some time but the attemptsto build analysis on this basis were not quite satisfactory. Among all such fields onehas to select those which in addition have more sophisticated properties. But for themoment let us stay with the Axiom in this preliminary form and see what we can dowith it.

The elements of R are called standard real numbers, while the elements of �Rnot in R are nonstandard. This terminology is taken from model theory but I findit not very suggestive in the present context. Sometimes the elements of �R arecalled hyperreal numbers. Perhaps someone some time will find a more intuitiveterminology.

The finite elements ˛ in �R are those which are not infinitely large, i.e. whichsatisfy Archimedes’ axiom: j˛j < n for some n 2 N (depending on ˛). These finiteelements form a subring E � �R. It contains all infinitesimal elements " which aredefined by the property that j"j < 1

nfor all n 2 N. It follows from the definition that

the set of infinitesimals is an ideal I � E. We have:

! infinitely large ” !�1 infinitesimal ¤ 0:

It is well known that this property characterizes E as a valuation ring in the sense ofKrull.

Theorem. The finite elements E form a valuation ring of �R with theinfinitesimals I as its maximal ideal. The residue class field E=I D R.

Two finite elements ˛; ˇ are said to be infinitely close to each other if ˛ � ˇ

is infinitesimal, i.e., if they belong to the same residue class modulo the ideal I ofinfinitesimals. This is written as

˛ � ˇ:

The residue class of ˛ 2 E is called the monad of ˛; this terminology has beenintroduced by Robinson in reference to Leibniz’ theory of monads. Every monadcontains exactly one standard number a 2 R; this is called the standard part of ˛,and denoted by st.˛/. There results the standard part map

st W E ! R


which in fact is nothing else than the residue class map of E modulo its maximalideal I .

In this situation let us consider the example of a parabola

y D x2

which, as I have narrated above, had been used by my school teacher to introduceus to analysis. Suppose x is a standard number. If we add to x some infinitesimaldx ¤ 0 then the ordinate of the corresponding point on the parabola will be

y C dy D .x C dx/2 D x2 C 2xdx C .dx/2

which differs from y by

dy D 2xdx C .dx/2 D .2x C dx/dx

so that the slope of the corresponding secant is

dy

dxD 2x C dx � 2x

since dx � 0 is infinitesimal. Hence:

If we step down from the hyperreal world into the real world again by using thestandard part operator, then the secant of two infinitely close points becomes thetangent, and the slope of this tangent is the standard part: st

�dydx

� D 2x.

I believe that such kind of argument had been used by my school teacher asnarrated above. As we see, this is completely legitimate.

It is apparent that in the same way one can differentiate any power xn instead ofx2, and also polynomials and quotients of polynomials, i.e., rational functions, withcoefficients in R. All the well known algebraic rules for derivations can be obtainedin this way. However, analysis does not deal with rational functions only. What canbe done to include more functions?

6.2.2 The Axiom in its final form

As described by the preliminary Axiom, �R is an ordered field. This can be expressedby saying that �R is a model of the theory of ordered fields. The theory of orderedfields contains in its vocabulary the function symbols “C” for addition, and “�” formultiplication, as well as the relation symbol “<” for the ordering. The axioms ofordered fields are formulated in this language. If we add to the vocabulary constantsfor all real numbers r 2 R and to the theory all statements which are true in R thenthe models of this theory are precisely the ordered field extensions of R.

If we wish to talk about functions and relations which are not expressible in thislanguage, then we have to use a language with a more extended vocabulary. In order


not to miss anything which may be of interest let us include into our language symbolsfor all relations in R. 1 The theory of R consists of all statements in this languagewhich hold in R. Thus, if we generalize the first part of the above Axiom as

�R is a model of the theory of R,

then this will allow us to talk in �R about every function and relation which is definedin R.

In order to generalize the second part of the Axiom we have to refer not onlyto the relation “<” of the ordering, but to every relation of similar kind. Moreprecisely: Consider a 2-place relation '.x; y/ defined in R. Such a relation is said tobe concurrent if, given finitely many elements a1; : : : ; an 2 R in the left domain of', there exists b 2 R in its right domain such that '.ai ; b/ holds for i D 1; : : : ; n. 2

Such element b may be called a “bound” for a1; : : : ; an with respect to the relation '.

Axiom (final form).

(1) �R is a model of the theory of R.

(2) Every concurrent relation' over R admits a universal boundˇ 2 �R,i.e., such that '.a; ˇ/ holds simultaneously for all a 2 R which arecontained in the left domain of '.

It is clear that this form of the Axiom is a generalization of its preliminary form,and a far reaching generalization at that. It was Abraham Robinson who had observedthat Leibniz, when he worked with infinitesimals, seemed tacitly to use somethingwhich is equivalent to that Axiom.

Of course, the essential point is that indeed there exists a structure �R satisfyingthis Axiom. This is guaranteed by general results of model theory. The most popularconstruction is by means of ultrapowers.

There is some ambiguity which has to be cleared. The Axiom refers to the “theoryof R”, and this refers to a given language as described above, its vocabulary includingsymbols for all relations over R. On first sight one would think of relations (andfunctions) between individuals, i.e., elements of R. This would lead to the firstorder language (Lower Predicate Calculus), where quantification is allowed overindividuals only. But in many mathematical investigations it is necessary to enlargethe language such as to contain also symbols for sets of functions, relations betweensets of functions etc., and quantification should be allowed over entities of any giventype. For instance, if we wish to state the induction axiom for the set N of naturalnumbers, we may say that

1Functions can be viewed as 2-place relations and thus are included. Subsets may be defined as the range oftheir characteristic functions and hence are included too.

2The relation ' may not be defined on the whole of R. The left domain of ' consists of those a 2 R forwhich there exists a b 2 R such that '.a; b/ holds. The right domain is defined similarly. For the orderingrelation “<” the left and the right domain coincides with R.


Every non-empty subset of N contains a smallest element

and this statement contains a quantifier for subsets.In order to include such statements too we have to work with the Higher Order

Language containing symbols also for higher entities, i.e., relations between sets,functions of relations between sets, etc. Quantification is allowed over entities of anygiven type. In other words:

We interpret the above Axiom as referring to the full structure over R and accord-ingly work with the corresponding higher order language.

This implies, among other things, that in �R we have to distinguish betweeninternal and external entities. Here we do not wish to go into details but refer, e.g.,to the beautiful introduction which Abraham Robinson himself has given in his bookNon-standard Analysis [Rob66]. See also the first section in [RR75].

Robinson introduced the terminology enlargement for a structure satisfying theAxiom. As said above, such an enlargement can be obtained by ultrapower construc-tion. It is not unique. In the following we choose one such enlargement and regardit as a fixed mathematical universe during our discussion.

6.2.3 Some excercises

Having learned all this from Abraham Robinson, my immediate reaction was whatprobably every newcomer would have done: I wished to put this method of reasoningto a test in simple exemplary situations. I do not have time here for a long discussionalthough much could be said to convince the reader of the enormous potential of thenew way of reasoning which Robinson’s theory of enlargements offers to us. Herelet me be content with a few examples.

Let f be a standard function and consider an element x 2 R in its domain ofdefinition. According to the part (1) of the Axiom, f extends uniquely to a functionon �R.

Continuity. f is continuous in x if and only if

x0 � x H) f .x0/ � f .x/: (6.1)

Of course, it is assumed that x0 is contained in the domain of f , so that f .x0/ isdefined. If the domain of the original f is open then f .x0/ is defined for every x0 inthe monad of x.

The above statement can be used as definition of continuity of a function. Notethat the usual quantifier prefix 8 " 9 ı : : : is missing.

I have chosen this example because I found precisely this definition in an oldtextbook. This was the German Kiepert, Differential- und Integralrechnung, of whichthe first edition had appeared in 1863. It had been very popular, and it got at least12 editions, the 12th appearing in 1912 [Kie12]. The text there reads as follows (inEnglish translation):


If some function is given by y D f .x/ then, in general, infinitely smallchanges of x will give rise to infinitely small changes of y. For all valuesof x where this is the case, the function is called continuous.

We see that this is precisely the definition (6.1).

Derivative. Let dx be an infinitesimal. Define dy by y C dy D f .x C dx/. Thenthe derivative f 0.x/ 2 R is defined by

f 0.x/ � dy

dx: (6.2)

More precisely: it is required that this is a valid definition, i.e., the quotient dydx

shouldbe finite and its monad should be independent of the choice of the infinitesimal dx. Ifthis is the case then f is called differentiable at x and f 0.x/ is defined as the standardpart of dy

dx.

I have chosen this example since it is the definition presented by my schoolteacher mentioned above. It is well possible that she had been trained using Kiepert’stextbook.

Integration. Suppose the function f .x/ is defined in the closed interval Œa; b� witha; b 2 R. Let n be a natural number and divide Œa; b� into n subintervals Œxi�1; xi � ofequal length. We take n infinitely large; then the length dx D b�a

nof each subinterval

is infinitesimal. Now the integral is defined by:

Z b

a

f .x/dx �nX

iD1

f .xi /dx: (6.3)

More precisely: It is required that this is a valid definition, i.e., the sum on the righthand side should be a finite element in �R and its monad should be independent ofthe choice of the infinite number n. If this is the case then f is called (Riemann)integrable over Œa; b� and the integral

R b

af .x/dx is defined as the standard part of

that sum.Maybe some explanation about infinite natural numbers is in order. �R is an

enlargement of R, and therefore every subset of R has an interpretation in �R. Sodoes N. This new subset of �R is denoted by �N. (In fact, �N is an enlargement ofN.) Using part (2) of the Axiom, it follows that there exists n 2 �N which is largerthan every number in N, i.e., n is infinite. The sum on the right hand side of (6.3)is to be interpreted as follows: For every finite n 2 N the sum sn D Pn

iD1 f .xi /dx

has finitely many terms, and so sn is well defined in R. The function n 7! sn from Nto R has an interpretation in the enlargement, i.e., it extends to a function from �N to�R. Thus sn is defined for every n 2 �N. Note that sn for infinite n is not an infiniteseries in the usual sense. It is to be regarded as the nonstandard interpretation of asum whose number of terms is a natural number.


The definition (6.3) of the integral explains Leibniz’ idea that the integral is es-sentially a sum (up to infinitesimals). This idea had led him to introduce the integralsign

Ras a variant of the letter S which he used for sums (instead of† which is used

today).I have been inspired to choose example (6.3) because of its relation toArchimedes’

method of measuring the area of a plane region. This method consists of cutting thearea into parallel strips of, say, length ` and infinitesimal breadth "; then ` � " is the(infinitesimal) area of the strip and the sum of all those areas will give the area of thewhole region – up to infinitesimals. The Leibniz formula (6.3) does precisely this inthe case of a positive function, when the region to be measured is that between thefunction graph and the x-axis.

That Archimedes had indeed used this idea (contrary to what is commonly at-tributed to him) is well documented by the Archimedes Codex which has been re-cently discovered and deciphered; see the report [NN07] about what is called “theworld greatest palimpsest”.

6.3 Robinson’s visits

6.3.1 Tübingen

As said at the beginning I had met Abraham Robinson in Los Angeles in Californiaduring my sabbatical. In the summer term of 1963 I was back at my universityin Tübingen. There I started a workshop where together with some students andcolleagues, we read Robinson’s papers and his book on model theory [Rob63] whichhad just appeared. We tried to understand his ideas for nonstandard analysis and toapply them to various situations. His book on nonstandard analysis [Rob66] had notyet been written.

Some time later when I had heard that Robinson was in Germany, I was able tomeet him and suggested that he spend a month or so in Tübingen as visiting professor,for a course on a topic from nonstandard analysis. He reacted favorably and so hevisited us in Tübingen in the summer of 1966.3

I had advertised his lecture course to students and colleagues, and so he had a fullauditorium. The aim of the course, two hours weekly, was to cover the fundamentalsof model theory with particular emphasis on the application to analysis and algebra.This job was not easy since the students (and most colleagues) did not have a formaltraining in mathematical logic; so he had to start from scratch. He was not what maybe called a brilliant lecturer who would be able to rouse a large audience regardless

3I am relying here on the extensive Robinson biography by Dauben [Dau95] where this year is recorded forRobinson’s Tübingen visit. Unfortunately I did not save our letters or other documents from that time and so Ihave to look at Dauben’s book for help in the matter of dates.


of the content of his talk. His way was quiet, with great patience when questionscame up from the students, but strong when it came to convince the students about theimpact of nonstandard applications. And this kept the attention of the large audiencethroughout his lecture.

In addition Robinson was available for discussion in our workshop. Just in timehis book on nonstandard analysis [Rob66] had appeared; he presented to us some ofthe more sophisticated applications.

I recall my impression that his Tübingen visit could be considered as a success,and from what is reported in Dauben’s biography it appears that Robinson thoughtso too.

6.3.2 Heidelberg

Next year, 1967, I moved from Tübingen to the University of Heidelberg. The generalacademic conditions in Heidelberg in those years were quite favorable. So it wasnot difficult to convince the faculty and the rector (president) that the visit of adistinguished scholar like Abraham Robinson would be of enormous importancefor the development of a strong mathematics group in Heidelberg. And so in thefollowing year, 1968, I was able to extend a cordial invitiation to Abraham Robinsonto visit us again, this time in Heidelberg. And he came, this time not from UCLA butfrom Yale where he had moved in the meantime.

Again he delivered a course on model theory and applications. To a certain extentthis job was kind of a repetition of his Tübingen lecture; again he had a large audience.But there was a difference. For in his seminar, on a smaller scale, he found an audiencewhich was highly motivated. On the one hand, there was a group of gifted studentsand postdocs who had also switched from Tübigen to Heidelberg and who had alreadyattended Robinson’s Tübingen lecture. On the other hand, in Heidelberg there hadbeen regular courses on Mathematical Logic (by Gert Müller who held a position as“associate professor”), and so there had been opportunities for the students to acquireknowledge in this field, in particular in model theory.

But the essential new feature of Robinson’s Heidelberg visit was that he talkednot only on nonstandard analysis but also on nonstandard algebra and arithmetic; inthe seminar he was able to expound his ideas in more detail. This found a respondentaudience. His impact on the work of these young people in the seminar was remark-able. And so it came about that he more or less regularly visited us in Heidelbergduring the following years, continuing his seminar talks and working with those whoresponded to his ideas.

In the next two sections I will give some kind of overview on the work resultingof his influence on the Heidelberg group, which was apparent even after his untimelydeath in April 1974.

Robinson’s influence was also helpful in another project. In view of Robinson’sstriking applications of model theory to mathematics proper, I became convinced that


a chair devoted to mathematical logic could be of help to mathematicians in theirdaily work, in particular if the chair was occupied by someone from model theory.Therefore I tried to obtain help from the university administration and the ministry ofeducation for establishing such a chair in the mathematics faculty. I had started thisproject in Tübingen already but after I moved to Heidelberg this would have to be achair for the Heidelberg faculty. Indeed, after some time such a chair was installed(in those times such thing was still possible). This was in 1973. It was clear to methat Robinson’s encouragement and judgement had been of great help in this matter.When I asked him whether he would accept an offer to Heidelberg for this chair thenhe did not say “no” but from the way he reacted it seemed to me that he really meant“no”. After all, Heidelberg seemed to be no match for Yale at that time. In any case,in a few months after that the problem was not existent any more. But it should beremembered that this chair, which is still in existence, had been installed with thestrong help of Abraham Robinson.

During his repeated visits to Heidelberg we came to know Abraham Robinsonnot only as a mathematician and scholar but also as a friend. He lived around thecorner from our house and on his way to town he regularly stepped in for a coffeeand conversation with us. (If I say “we” and “us” in this context then I include mywife Erika.) He was a man with a wide horizon and far reaching interests. If hetalked about Leibniz then one could feel not only his knowledge about Leibniz’ lifeand work but also his sympathy for that remarkable man. There was only one thingabout which he strongly disagreed with Leibniz, namely Leibniz’ insistence that “ourworld is the best of all possible worlds”.

Abby liked to talk to people, and sometimes we had the impression that he knewmore about our neighbors than we did. He was keenly interested in the local history.When we took him on tour to show him the country and its places then it often turnedout that he knew more about it than we did, and he gave us a lecture on the history ofthose places.

In the course of those years there developed a friendship of rare quality. Abbybelongs to the few close friends whom I have met in my life. I have learned much fromhim, not only in Mathematics but also in questions of attitude towards the problemsof life.

6.4 Nonstandard algebra

Looking at the Axiom in its final form (in section 6.2.2) it is apparent that this Axiomhas little to do with the special properties of the real number field R. It makes sensefor every mathematical structure. And so there is not only nonstandard analysis, butnonstandard mathematics at large. Abraham Robinson was well aware of this; hehas applied his method, partly in collaboration with others, to various mathemati-


cal problems ranging from topology, Hilbert spaces, Lie groups, complex analysis,differential algebra, quantum theory to mathematical economics.

There were also investigations in the direction of algebra and number theory. Assaid above, Abraham Robinson reported on this in his Heidelberg seminar lectures.One of his first topics was his nonstandard interpretation of Hilbert’s irreducibilitytheorem (jointly with Gilmore in [GR55] ). This paper of Robinson has been saidto mark a “watershed” in the development of model theory (in the same line withanother paper of Robinson’s, of the same year 1955, on Artin’s solutiom of Hilbert’s17th problem [Rob55] ).

Hilbert had published his irreducibility theorem in 1892 [Hil92]. Suppose thatf .X; Y / is an irreducible polynomial in 2 (or more) variables then, Hilbert showed,there are infinitely many specializationsX 7! t such thatf .t; Y / remains irreducible.The coefficients of f are taken from the rational field Q and the specialized variablet is also assumed to be in Q. Since then there had been numerous proofs of thistheorem, also over other base fields K, e.g., number fields. Hasse had the idea tostudy arbitrary fields over which Hilbert’s irreducibility theorem may hold, and hisPh.D. student Wolfgang Franz started the theory of such fields which today are calledHilbertian fields [Fra31]. This was the point where Abraham Robinson stepped in.He stated a nonstandard characterization of Hilbert fields.

As a follow-up of our discussions with Robinson we were able to amend his resultof [GR55] by presenting a new, “metamathematical” proof of Hilbert’s irreducibilitytheorem in the number field and the function field cases. It turned out that Hilbert’sirreducibility is, in fact, equivalent to the well known theorem of Bertini in algebraicgeometry [Roq75]. Further investigations by R. Weissauer showed that every fieldwith a set of valuations satisfying the product formula is Hilbertian. This covered allclassical fields which were known to be Hilbertian. Moreover, Weissauer found quitea number of new and interesting Hilbertian fields, e.g., formal power series fields inmore than one variable [Wei82].

Weissauer’s paper is a good example of the usefulness of Robinson’s enlargements.On the one hand, it can be shown that any result which has been proved using thenotion and the properties of enlargements can also be obtained without this. On theother hand, the use of enlargements provides the mathematician with new methods andit opens up new analogies to other problems which sometimes help to understand thesituation. Abraham Robinson used to say that his method may reduce a “dynamical” toa “statical” situation. For instance, an infinite sequence t1, t2, t3; : : : which preservesthe irreducibility of the polynomial f .X; Y / under the specialization X 7! ti leadsto a nonstandard t which renders f .t; Y / irreducible.

For another topic of algebra: Remember that group theory had been started byGalois in order to study the roots of algebraic equations. Today the notion of Galoisgroup belongs to the basics of algebra. But there arose the need to study simulta-neously infinitely many algebraic equations; this led Krull in 1928 to the discoveryof the topological structure of infinite Galois groups [Kru28], and this developed


into the theory of profinite groups. Robinson has pointed out that profinite Galoisgroups can be naturally understood within the enlargement, connected to the “finite”groups in the sense that their order is an infinitely large natural number n 2 �N. Thecorresponding profinite groups in the standard world are obtained from these non-standard “finite” groups by a similar process as the derivative f 0.x/ is obtained fromthe nonstandard differential quotient dy

dxin the manner as explained above. Hence

again:

If we step down from the nonstandard world into the standard world again, thenKrull’s Galois theory of infinite algebraic extensions appears as an immediate con-sequence of the Galois-Steinitz theory for finite algebraic field extensions.

There arises the interesting question which fields K are uniquely determined (upto elementary equivalence) be their full profinite Galois groups GK . See [Pop88],[Koe95].

The description of the structure ofGQ as profinite group is at the focus of currentarithmetical research.

6.5 Nonstandard arithmetic

Remember Hensel’s p-adic number fields which Hensel had conceived at around1900 and which today have become standard tools in algebraic number theory andbeyond. In the course of time it became necessary to consider all p-adic completionsat once; this has led to the introduction of adeles and ideles in the sense of Chevalleywhich play a fundamental role, e.g., in class field theory. Now, Abraham Robinsonhas pointed out that his notion of enlargement comprises all those constructions at thesame time. His enlargements are indeed the most universal “completions” in as muchas every concurrent relation admits a bound. The classical notions of p-adics, adelesand ideles, profinite groups etc. are obtained from his enlargement by a universaltransfer principle, similar to obtaining the derivative f 0.x/ as the standard part of thedifferential quotient dy

dxas explained above.

In the ensuing discussions with Abraham Robinson we wished to test his methodin some more situations of fundamental importance. The Siegel–Mahler Theoremseemed to us a good example to begin with. Finally in November 1973 he invitedme to Yale with the aim of discussing in more detail the possibility of a nonstandardproof of this theorem.

Let �: f .x; y/ D 0 be an irreducible curve defined over a number field K offinite degree. If � is of genus g > 0 then Siegel’s theorem says that � admits onlyfinitely many points whose coordinates are integers in K. Mahler had generalizedthis by proving that for any finite set S of primes of K there are only finitely manypoints in � whose coordinates are S -integers inK. The S -integers are those numbersin K whose denominator consists of primes in S only.


Nonstandard methods seem to be useful to distinguish between finite and infinite.We work in a fixed enlargement �K ofK, with the properties as statetd in the Axiom.If � would admit infinitely many S -integral points in K then it would also admit anonstandard S -integral point in �K. Such a point .x; y/ is a generic point of � overK and hence F D K.x; y/ is the function field of � over K. By construction F isembedded into �K:

F � �K:

Now, both these fields carry a natural arithmetic structure: F as an algebraic functionfield over K and �K as a nonstandard model of the number field K. What is therelation between the arithmetic in F and in K? In our joint paper [RR75] we wereable to prove the following

Theorem 1. If F is of genus g > 0 then every functional prime divisorP of F is induced by some nonstandard prime divisor p of �K.

From here it is only a small step to deduce the validity of the Siegel–MahlerTheorem. Abby agreed to work out the proof of the theorem for elliptic curves, and Iwas to deal with curves of higher genus. Two weeks after I had left Yale he sent mehis manuscript for the elliptic part. But he could not see any more my part for highergenus.

Actually, there is a famous conjecture of Mordell to the effect that a curve �of genus g > 1 has only finitely many K-rational points, even without specifyingthat they are S -integers. This conjecture has been proved by Faltings in 1983. Innonstandard terms it can be formulated as follows:

Theorem 2. A function field F jK of genus g > 1 cannot be embeddedinto the enlargement �K.

Clearly, this contains Theorem 1 in the caseg > 1, which was my own contributionin the joint work with Robinson. But in 1973 Mordell’s conjecture had not yet beenproved and hence, at that time, the proof of Theorem 1 was necessary also for thecase g > 1.

In 1973 I discussed with Robinson also a possible nonstandard proof of Mordell’sconjecture. We planned first to develop the tools which we believed to be necessaryfor this project. However, due to Robinson’s sudden death our plan could not berealized.

In later years Kani [Kan80b], [Kan80a], [Kan82] has studied systematically func-tion fields which are embedded into the enlargement �K. In my opinion, the toolsand the results which have been obtained in his work are well capable to give a non-standard proof of Mordell’s conjecture, together with Roth’s theorem (which is alsoused in the proofs of Theorems 1 and 2). But this has not yet been worked out. Itremains an open challenge.

Chapter 7

Cahit Arf and his invariant

Cahif Arf and his invariant

by Falko Lorenz and Peter Roquette

Mitteilungen der Mathematischen Gesellschaft in Hamburg Bd. XXX (2011), 87–126.

7.1 Introduction 1907.2 Arf’s first letter 1917.3 Some personal data 1947.4 Quadratic spaces 1967.5 Clifford algebras 1987.6 Binary quadratic space 1997.7 Higher dimensional quadratic spaces 2037.8 Witt equivalence 2067.9 Arf’s Theorems 2077.10 An assessment of Arf’s paper 2127.11 Perfect base fields 2147.12 Epilog 2157.13 Appendix: Proofs 216

Originally this manuscript was prepared for my talk at the workshop on Sequences, Curvesand Codes in Antalya, 25–29 September 2009. Later I had given a talk with the same title onOctober 5, 2009 at the conference on Positivity, Valuations, and Quadratic Forms in Konstanz.

In the discussion after the talk I learned that there may be an error in Arf’s paper andperhaps his main theorem has to be modified. I am indebted to Karim Johannes Becher for thiscomment. Indeed, after another check I found the error in the proof of one of Arf’s lemmas.Accordingly the manuscript had to be corrected, taking care of the situation and clarifying thescope of Arf’s theorems after the correction.1

The present article is the result of unifying and re-editing our two papers mentioned above.For the convenience of the reader we have included an appendix containing very simple proofsof Arf’s main results in the corrected form. It seems to be of interest that all the facts which weuse in these proofs can be found in Arf’s paper already.

1We are indebted to Detlev Hoffmann for providing us with the relevant information. Also we would like tothank K. Conrad and S. Garibaldi for their help in this matter. We are indebted to the referee for his informativecomments, who called our attention to the book [EKM08], in particular §39.

190 7 Cahit Arf and his invariant

7.1 Introduction

In January 2009 I received a letter from the organizers of the workshop in Antalyaon Sequences, Curves and Codes, with a friendly invitation to participate. The letterwas accompanied by a bank note of 10 Turkish Lira. Reading the letter I found outthat this was not meant as an advance honorarium for my talk, but it was to tell methat the note carried the portrait of the Turkish mathematician Cahit Arf (1910–1997).Besides the portrait there appears some mathematical text pointing to Arf’s discoveryof what today is called the Arf invariant. Accordingly the organizers in their lettersuggested that perhaps I would want to talk about the Arf invariant of quadratic forms.

Cahit Arf on Turkish bank note

Although I do not consider myself as a specialist on quadratic forms, it was mypleasure to follow this suggestion. CahitArf had been a Ph.D. student of Helmut Hassein 1937/38. Arf’s thesis [Arf39] has become widely known, where he had obtaineda generalization of a former theorem of Hasse about the ramification behavior ofabelian number fields; today this is known as the “Hasse–Arf theorem”.2 His nextpaper, after his thesis, contains the “Arf invariant” which is our concern here. Thiswork too was inspired by a suggestion of Hasse. So this report about the Arf invariantfits into my general project to investigate the mathematical contacts of Hasse withvarious other mathematicians, including Emil Artin, Emmy Noether, Richard Brauerand others, and now with Cahit Arf.

Much of what I know in this respect is based upon the letters between Hasse andhis correspondence partners. Those letters are kept in the Handschriftenabteilungof the Göttingen University Library, they contain a rich source for those who areinterested in the development of algebraic number theory in the 20th century. Amongthose documents there are preserved about 65 letters between Arf and Hasse from1939 until 1975.3 We can see from them that in the course of time there developed aheartfelt friendship between the two.

2This generalization had been asked for by Artin in a letter to Hasse. For details from the historic perspectivesee, e.g., section 6 of [Roq00].

3In addition there are about 90 letters between Hasse and Arf’s wife Halide, mostly in Turkish language, inwhich Hasse tried to practice and improve his mastery of the Turkish language.


7.2 Arf’s first letter

The first ten letters are concerned with Arf’s work on quadratic forms in characteris-tic 2. But where are the earlier letters, those about Arf’s thesis? The answer is easy:There were no earlier letters, for during his graduate studies while composing histhesis, Arf worked at Göttingen University where Hasse was teaching. And peopleat the same university usually do not write letters but talk to each other.

Fortunately for us, when Arf worked on quadratic forms in characteristic 2 he wasback in Istanbul, and therefore the communication with his former academic teachertravelled by means of letters. On October 12, 1939 Arf wrote to Hasse:4

Sehr geehrter Herr Professor,

Ich habe Ihren Brief vom 29. 9. 39 mit grosser Freude erhalten …Ich habe jetzt eine unschöne Arbeit über quadratische Formen fast fertiggeschrieben. Diese Arbeit wollte ich Ihnen vorlegen. Ich glaube aber,dass Sie jetzt wenig Zeit haben. Es handelt sich kurz um folgendes:

Sie hatten einmal den Wunsch geäußert, die Geschlechtsinvarianten ein-er quadratischen Form mit Hilfe der Algebrentheorie begründet zu sehen.Ich habe versucht dies zu tun. Da die Aufstellung dieser Invarianten fürp ¤ 2 fast trivial ist, habe ich gedacht, dass es nützlich sein würdewenn man zunächst die Theorie in Körpern von der Charakteristik 2 zuübertragen versucht. In der genannten Arbeit übertrage ich die Ergeb-nisse von Witt durch passende Änderungen in den Körpern von derCharakteristik 2 und ich gebe dann die vollständigen Invariantensys-teme für arithmetische Äquivalenz der ternären und quaternären Formenin einem Potenzreihenkörper k..t//, wobei der Koeffizientenkörper kdie Charakteristik 2 hat und vollkommen ist …

Dear Professor,

I am very glad to have received your letter of September 29, 1939 … Ihave almost completed the draft of a paper on quadratic forms. I hadintended to submit it to you. But I believe that now you will not havemuch time for it. In short, the situation is as follows:

You had once expressed your wish to see the genus invariants of aquadratic form be established with the help of the theory of algebras.This I have tried to do. Since the compilation of those invariants is al-most trivial in characterisctic ¤ 2 I thought it would be useful at firstto try to transfer the theory to fields of characteristic 2. In the abovementioned paper I transfer the results of Witt by suitable modifications

4Observe the date of this letter. On September 1, 1939 World War II had started. Perhaps this was the reasonwhy Arf believed that Hasse would not be able to devote much time to deal with Arf’s paper since in war timeHasse may have been assigned to other duties.


to fields of characteristic 2. And then I give a complete system of invari-ants for the arithmetic equivalence of ternary and quaternary forms in apower series field k..t//5 where the field of coefficients k is perfect ofcharacteristic 2 …

From this we learn that it had been Hasse who had suggested the topic of Arf’sinvestigation. Hasse’s interest in quadratic forms stems from the time of his ownthesis, 1923/24, when he had proved the Local-Global Principle for quadratic formsover number fields [Has23a, Has24a]. Later he had established the Local-GlobalPrinciple for central simple algebras over number fields, in cooperation with EmmyNoether and Richard Brauer [BHN32]. There arose the question as to the mutualinterrelation between the theory of quadratic forms and the theory of algebras. Perhapsit would be possible to deduce the Local-Global Principle for quadratic forms fromthat for algebras ?

This question (and more) had been answered beautifully in Witt’s seminal paper[Wit37a]. At that time Witt held the position as assistant professor in Göttingen, andhe was the leading member of the Arbeitsgemeinschaft in cooperation with Hasse. Inhis paper Witt associates to every quadratic form f a central simple algebra S.f / of2-power index, called the Hasse algebra which, together with the dimension and thediscriminant of the form, makes a complete set of invariants at least over global andlocal function fields. (Over number fields there is an additional invariant, namely thesignature of a quadratic form over the real localizations of the field.)

Witt’s paper represents a watershed in the theory of quadratic forms. It providedthe basis of the subsequent enormous expansion of the theory of quadratic forms. Hisbiographer Ina Kersten says that this paper “ranks as one of his most famous works”[Ker00]. However, Witt’s theory covered only forms over a field of characteristic ¤ 2.This is the point where Arf’s paper comes in. He extended Witt’s theory to fields ofcharacteristic 2. In particular this applies to the case of local and global functionfields of characteristic 2.

The desire to extend Witt’s result to characteristic 2 had also been expressed byA.A.Albert in a paper which had just appeared in 1938 in the Annals of Mathematics[Alb38b]. There had been some letters exchanged between Albert and Hasse duringthe years 1931–1935 and we know that Albert’s interest in the theory of quadraticforms over global fields had been encouraged by Hasse.6 The above mentionedpaper by Albert shows that this interest continued. His paper is a follow-up onWitt’s [Wit37a] on quadratic forms: Albert first reproves, in his own way, some ofWitt’s results for global function fields and then shows that these hold also in thecharacteristic 2 case: namely that every quadratic form in 5 or more variables isisotropic. And then, in a footnote, he says about Witt’s general theory:

5Arf writes here �hti but we will use throughout our own notation and do not follow the various notationsin the original letters and papers.

6See chapter 8 in [Roq05].


The results of Witt on quadratic forms on a field of characteristic nottwo may probably be obtained for the characteristic two case only forforms with cross product terms.7 It would be very interesting to studythe analogues of Witt’s results for our characteristic two case but theauthor has not yet done so.

We see that Arf did just what was proposed here. I do not know whether Arf knewabout Albert’s paper and its footnote. In his own paper he cites Witt only and says inthe introduction:

Die Anregung zu dieser Arbeit verdanke ich H. Hasse.

I owe to H. Hasse the suggestion for this work.

As we learn from Arf’s letter, there was a second part in his manuscript where heinvestigates quadratic forms over rings of power series and their arithmetic invariants,at least for quadratic forms of low dimension. Again, the motivation for this comesfrom number theory. Due to his results in this second part Arf can be regarded as aforerunner of the general theory of quadratic forms over rings, not necessarily fields.8

But in the end it turned out that Parts 1 and 2 were published separately. Part 1appeared in Crelle’s Journal where Hasse was editor [Arf41]. Hasse would haveliked to get also Part 2 for Crelle’s Journal but it seems that there arose difficultieswith the printing due to paper shortage in war times and so the second part appearedin the journal of the University of Istanbul.9

Here I will discuss only the first paper of Arf [Arf41] where he introduces his Arfinvariant. It has turned out that there is an error in Arf’s paper which on the one handreduces the scope of his main result but on the other hand has led to an interestingdevelopment in the theory of quaternion algebras over fields of characteristic 2. Weshall discuss this in due course.

But before going into details let us familiarize a little with the people involvedand with the time of the game.

7It appears that Albert means quadratic forms which are “regular” in the sense as defined below (and “com-pletely regular” in Arf’s terminology). See Section 7.4.

8This is evident when we look at the book by Knus who gives a survey on quadratic forms over rings [Knu80]where the notion of Arf invariant over rings is systematically treated. (However we do not find Arf’s Part 2[Arf43] cited in the bibliography of Knus’ book.) It seems that the most recent survey on the topic is containedin the book [EKM08].

9See [Arf43]. The paper is usually cited for the year 1943 but at the end of the paper we read that it has beenreceived on May 15, 1944.


7.3 Some personal data

Cahit Arf was born in 1910 in the town of Selanik which today is Thessaloniki. Atthat time it belonged to the Ottoman empire. But in the course of the Balkan war1912 his home town was affected and the family escaped to Istanbul, later in 1919 toAnkara and finally moved to Izmir.10

Cahit Arf’s childhood encompassed the Balkan wars, the World War I,the grand war at Gallipoli, the Greek invasion of westernAnatolia and theinvasion of Istanbul by the Allied Powers. When finally Turkey emergedas a new independent parliamentary republic in 1923 Cahit Arf was 13years old. It was the beginning of a new era. The new Republic washopeful, determined and full of invincible self confidence. These traitswere also deeply trenched in young Arf. This would shape his attitudetowards mathematics in the future.

In public school Arf’s ease in mathematics was soon to be noticed by his teachers.In 1926 his father sent him to France to finish his secondary education at the prestigiousSt. Louis Lycée. Because of his extraordinary grades in mathematics he graduated intwo years instead of the expected three years. Then he obtained a state scholarshipto continue his studies at the École Normale Supérieur, again in France.

After his return to Turkey in 1932 Arf taught at high school and since 1933 heworked as instructor at Istanbul University.

It did not take him long to realize that he needed graduate study inmathematics. In 1937 he arrived at Göttingen University to study withHelmut Hasse.

At that timeArf was 27 years of age. I do not know why he had chosen Göttingen ashis place of graduate study. Although Göttingen used to be an excellent mathematicalcenter which was attractive to students throughout the world, that period had endedin 1933 when the new Nazi government decided to discharge the Jewish and thenon-conformist professors; this had disastrous effects to the mathematical scene inGöttingen. It seems improbable that Arf had not heard about the political situationin Germany and its consequences for the academic life in Göttingen. Perhaps hismathematical interests at that time leaned towards algebra and arithmetics and he hadfound out (maybe someone had advised him) that in Göttingen there was Hasse whowas known as an outstanding mathematician in those fields. In fact, measured by thehigh standard of Arf’s thesis which he completed within one year after his appearancein Göttingen, it seems that already in Istanbul he had acquired a profound knowledgein the basics of modern algebra and algebraic number theory, and accordingly he mayhave chosen Göttingen because of Hasse’s presence there.

10The following citations are taken from the Arf biography written by Ali Sinan Sertöz [Ser08]. There onecan find more interesting information about the life, the work and the personality of Cahit Arf.


Helmut Hasse was 38 years of age when in 1937 Arf arrived in Göttingen. Hassewas known as a leading figure in the development of algebraic number theory, inparticular of class field theory. Just one year earlier in 1936 he had been chosen asan invited speaker at the International Congress of Mathematicians in Oslo. Therehe reported on his proof of the Riemann hypothesis for elliptic function fields overfinite fields of constants; this proof had appeared in three parts in the 1936 issue ofCrelle’s Journal [Has36c], [Has36b], [Has36a]. I would say that in those years Hassewas at the height of his mathematical power (notwithstanding a certain peak of hismathematical activities in the years after World War II). In 1934 Hasse had decided toleave the University of Marburg and to accept an offer to Göttingen. Hasse was not aNazi but he described his political position as being patriotic. He strongly disagreedwith the policy of expelling so many scientists from Germany; he considered this asa tragic loss of intellectual power in Germany. And he tried to do what he could tocounteract this. When he decided to move from Marburg to Göttingen he did this withthe expressed intention to restore, at least to a certain extent, the glory of Göttingenas an international place for mathematics. Although he spent a lot of time and energyon this, he could not be successful in the political situation.11

However, on a relatively small scale Hasse’s activity in Göttingen had remarkablesuccess. He managed to attract a number of highly motivated students to his seminarand the Arbeitsgemeinschaft. The latter was organized by Witt but Hasse participatedat the meetings and led the mathematical direction of the work.

The high scientific level of the work in the Arbeitsgemeinschaft is documented ina number of publications in Crelle’s Journal and other mathematical journals. Herewe only mention volume 176 of Crelle’s Journal which appeared just in 1937 whenArf came to Göttingen. A whole part of this volume12 contains papers which arosein the Arbeitsgemeinschaft and in the Seminar, of which Witt’s famous paper on theso-called Witt vectors is to be regarded as a highlight. In the same volume (but inanother part) appeared Witt’s paper on quadratic forms [Wit37a] which, as said earlieralready, has decisively influenced Arf’s paper on his invariant [Arf41].

Ernst Witt was 26 when he met Arf, hence one year younger. He had studiedin Göttingen since 1930. He had received the topic of his Ph.D. thesis from EmmyNoether but since she had been dismissed she could not act as his thesis referee.13

The thesis was concerned with central simple algebras over function fields in thecourse of which he proved the Riemann-Roch theorem for algebras, a ground breakingpaper [Wit34] which nowadays attracts new interest in the setting of non-commutativealgebraic geometry.14 When Hasse came to Göttingen in 1934 he accepted Witt as hisassistant on the recommendation of Emmy Noether. Witt has not many publications

11For more facts from Hasse’s biography see, e.g., Frei’s biography [Fre85], as well as Frei’s recollectionsabout Hasse in [FR08]. Hasse’s involvement with the Nazi regime is discussed, e.g., in [Seg03].

12Each volume of Crelle’s Journal appeared in 4 parts (“4 Hefte”).13This was Herglotz with whom Witt maintained close relationship.14It is said that Witt completed the manuscript of his thesis within one week.


when compared to the work of other mathematicians, but each one is of high level andwitnesses a profound insight into matematical structure. We have already mentionedhis 1937 paper where he introduces “Witt vectors” [Wit36].15 Earlier the same yearthere had appeared his paper on quadratic forms [Wit37a] which actually constitutedhis Habilitation thesis. 16

It seems fortunate that Arf in Göttingen had the chance to join the inspiring andmotivated group of young mathematicians around Hasse, and among themWitt. Therearose a friendship between the two which lasted for many years. It is without doubtthat Arf’s paper on quadratic forms in characteristic 2 has been influenced by Witt’sin characteristic ¤ 2.

7.4 Quadratic spaces

Let K be a field. Classically, a quadratic form over K is given by an expression

q.x/ DX

1�i�j �n

aijxixj with aij 2 K: (7.1)

Two quadratic forms are said to be equivalent if one is obtained from the other bya non-degenerate K-linear transformation of the variables x D .x1; : : : ; xn/. Aninvariant is a mathematical entity attached to quadratic forms which does not changeif a form is replaced by an equivalent form.

Witt had replaced the above notion of quadratic form by the notion of quadraticspace which was adapted to the “Modern Algebra” of the time [Wit37a]. A quadraticspace overK is a vector space V equipped with a function q W V ! K and a bilinearfunction ˇ W V V ! K subject to the following conditions:

q.x/ D 2q.x/

q.x C y/ D q.x/C q.y/C ˇ.x; y/

μfor 2 K; x; y 2 V: (7.2)

We assume V to be of finite dimension n. If u1; : : : ; un is a basis of V then anyx 2 V may be written as x D x1u1 C � � � xnun with xi 2 K and then q.x/ appearsin the form (7.1) with aij D ˇ.ui ; uj / for i < j and ai i D q.ui /. In Witt’s setupthe notion of “invariant” now refers to isomorphisms of quadratic spaces instead ofequivalences of quadratic forms.

15It should not be forgotten that Witt vectors had been discovered somewhat earlier already by H. L Schmid[Sch35a] who also was one of Hasse’s assistants at that time. H. L. Schmid however worked with the main vectorcomponents only (Hauptkomponenten) where the formulas for addition and multiplication are quite cumbersome.It was Witt who observed that the structural operations for Witt vectors can be described quite easily in terms ofthe ghost components (Nebenkomponenten). In this way he made the calculus of Witt vectors widely applicable.

16More biographic information about Witt can be obtained from Ina Kersten’s biography [Ker00] and thearticles cited there.


It is common to interpret q.x/ as the “length” of the vector x 2 V , more preciselyas the square of its length. In fact, Witt and also Arf write jxj2 instead of q.x/.Similarly ˇ.x; y/ is interpreted as the “inner product” of the vectors x and y andaccordingly Arf writes x � y for it. Witt however writes x � y for 1

2ˇ.x; y/ which is

possible in characteristic ¤ 2 and corresponds more to our geometric intuition, forthen one has x � x D jxj2. In characteristic 2 however this is not possible and so wehave x � x D 2jxj2 D 0. In other words, in characteristic 2 we have to live with thefact that every vector is orthogonal to itself.

This has consequences. The first observation is that the process of diagonalizationis not generally possible in characteristic 2. Recall that in characteristic ¤ 2 everyquadratic form admits an equivalent “diagonal” form:

q.x/ DX

1�i�n

aix2i with ai 2 K: (7.3)

In Witt’s terminology this means that every quadratic space V admits an orthogonalbasis u1; : : : ; un, where q.ui / D ai . Thus V splits as an orthogonal direct sum ofone-dimensional quadratic subspaces:

V D ?X

1�i�n

Kui in characteristic ¤ 2. (7.4)

But in characteristic 2 this is not always possible. Arf observed that one has to admitalso two-dimensional subspaces:

V D ?X

1�i�r

.Kui CKvi /C ?X

1�j �s

Kwj in characteristic 2 (7.5)

where ui and vi are not orthogonal to each other, i.e., ˇ.ui ; vi / ¤ 0: After suitablenormalization we may assume that

ˇ.ui ; vi / D 1:17

The dimension n of V is n D 2r C s. Thus in characteristic 2 any quadratic formadmits an equivalent form as follows:

q.x/ DX

1�i�r

�aix

2i C xiyi C biy

2i

� CX

1�j �s

cj z2j (7.6)

forx D

X1�i�r

.xiui C yivi / CX

1�j �s

zjwj ;

where we have put

ai D q.ui /; bi D q.vi /; cj D q.wj /:

17Arf however does not assume this and he admits for ˇ.ui ; vi / any non-zero element in K. Therefore hisformulas for the Arf invariant look a little more complicated than ours.


Arf speaks of “quasi-diagonalization” since only the second sum in (7.6) is in pure“diagonal form” whereas the matrix of the first sum splits into 22 submatrices alongthe diagonal. Note that in characteristic 2 the square operator is additive; accordinglythe second sum in (7.6) is called the quasi-linear part of q. The second sum in (7.5)consists of all z 2 V which are orthogonal to V ; therefore it is denoted by V ?.

In characteristic ¤ 2 a quadratic form (7.1) is called regular, or equivalentlynon-singular, if it has no equivalent form which can be written in fewer than nvariables; this means that in the diagonal form (7.3) all the coefficients ai ¤ 0. Incharacteristic 2 things are different. Arf retained the above definition of “regular”; incharacteristic 2 this means that the coefficients cj in the quasi-linear part of (7.6) arelinearly independent over the subfieldK2. If there does not appear a quasi-linear part,i.e., if s D 0, then Arf called the quadratic form “completely regular” (“vollregulär”).

Today the terminology has changed. Instead of Arf’s “completely regular” onesays “regular” (or “non-singular”) whereasArf’s “regular” is now referred to as “semi-regular”. This reflects the experience that in the new terminology, the “regular” formsin characteristic2behave in many respects similar to the regular forms in characteristic¤ 2.

We shall use here the terminology of today, deviating from Arf’s.The notions of “regular” und “semi-regular” are invariant, and can also be applied

to a quadratic space: V is “regular” if V ? D 0, and V is “semi-regular” if V ? doesnot contain a vector x ¤ 0 with q.x/ D 0, i.e., if V ? is anisotropic.

Every quadratic space .V; q/ can be scaled. If 0 ¤ c 2 K then the scaled quadraticspace with scaling factor c is defined to be .V; cq/. Notation: V .c/. Thus in V .c/ the“length” of every vector x is c � q.x/ whereas in V it is q.x/.

7.5 Clifford algebras

The Clifford algebra C.V / of a quadratic space V is defined as an associativeK-algebra (not commutative in general) generated by the K-module V and withthe defining relations:

x2 D q.x/ for x 2 V: (7.7)

In view of (7.2) this implies

xy C yx D ˇ.x; y/ for x; y 2 V: (7.8)

If u1; : : : ; un is a K-basis of V then a K-basis of C.V / is given by the productsui1ui2 � � �uik with i1 < i2 < � � � < ik and 0 � k � n. The K-dimension of C.V /is 2n.

In view of its definition C.V / is an invariant of V . So is the subalgebra C0.V / �C.V / which is generated by the products ui1ui2 � � �uik with an even number k of


factors. The invariance of C0.V / is a consequence of the fact that the definingrelations (7.7) are of degree 2. The dimension of C0.V / is 2n�1.

If K is of characteristic 2 then we have the following rule:

If V D V1 ? V2 then C.V / D C.V1/˝ C.V2/: (7.9)

Here I have written V1 ? V2 to indicate the orthogonal direct sum of V1 and V2. Thetensor product is taken over K as the base field. The validity of (7.9) is immediateif we observe that any x1 2 V1 and x2 2 V2 are orthogonal to each other, i.e.,ˇ.x1; x2/ D 0. Hence from (7.8) we conclude that x1x2 D �x2x1 D x2x1 showingthat C.V1/ and C.V2/ commute elementwise. (At the same time we see that incharacteristic ¤ 2 this is not the case since the appearing minus sign cannot bedisregarded.)

In view of the decomposition (7.5) we obtain for characteristic 2 that the Cliffordalgebra C.V / decomposes into the tensor product of r factors C.Kui C Kvi / ofdimension 4, and the factor C.V ?/ of dimension 2s . The latter is the center of C.V /and does not appear if V is regular.

7.6 Binary quadratic space

First Arf investigates the Clifford algebra of a regular quadratic space which is binary,i.e., of dimension 2. This discussion is quite elementary but it is fundamental for allof Arf’s results.

A binary space is generated by 2 elements u; v with the relations

q.u/ D a; q.v/ D b; ˇ.u; v/ D 1: (7.10)

Its Clifford algebra C.V / is given by the defining relations

u2 D a; v2 D b; uv C vu D 1: (7.11)

This is a central simple algebra of dimension 4, i.e., a quaternion algebra, with theK-basis 1, u, v, uv.

The even subalgebra C0.V / is of dimension 2 and has the basis elements 1 anduv with the relation:

.uv/2 C uv D uv.vuC 1/C uv D u2v2 D ab:

Putting w D uv and introducing the Artin–Schreier operator }.X/ D X2 C X wemay write this as:

C0.V / D K.w/ with }.w/ D ab: (7.12)


Suppose first that ab … }.K/. Then K.w/ is a separable quadratic field exten-sion. According to the Artin–Schreier theory this field is uniquely determined by theresidue class of ab modulo }.K/. The nontrivial automorphism ofK.w/ is given byconjugation with u:

u�1wu D vu D w C 1: (7.13)

Since u2 D a we see that C.V / is a cyclic crossed product of the separable quadraticfield K.w/, whose factor system is determined by the element a 2 K� modulo thenorm group from K.w/�.

The original quadratic form q.x; y/ can be rediscovered (up to equivalence) fromthese data by the norm function N W K.w/ ! K as follows:

a �N.x1 C x2w/ D a � .x1 C x2w/.x1 C x2.w C 1//

D ax21 C x1.ax2/C b.ax2/

2 (7.14)

D q.x1; ax2/:

Although this is not the original quadratic form q.x1; x2/, it is equivalent to it. Thisformula can be interpreted as follows: We may regardK.w/ as a quadratic space withrespect to the norm function N W K.w/ ! K. Consider the scaled space K.w/.a/

which has the quadratic form aN W K.w/ ! K. Then formula (7.14) says thatK.w/.a/ as a quadratic space is isomorphic to V . This isomorphsm is given by1 7! u, w 7! a�1v.

As a consequence we state the following fact for later use:

The image set q.V / of the quadratic space V equals the coset of thenorms from K.w/ which contains a, i.e.:

q.V / D a �N.K.w//: (7.15)

In the above discussion we had assumed that ab … }.K/. The case ab 2 }.K/

is somewhat exceptional since in this case K.w/ is not a field but a commutativeseparable K-algebra which decomposes into the direct product of two copies of K.Writing ab D }.c/ with c 2 K and putting e1 D wC c and e2 D e1 C 1, we obtainorthogonal idempotents e1, e2, henceK.w/ D Ke1 ˚Ke2. By suitable choice of thebasis u, v of V one can achieve that u2 D a ¤ 0; then u admits an inverse in A andconjugation with u induces the automorphism of K.w/ which permutes e1 and e2.Hence again, C.V / is a crossed product of K.w/, determined by the element a 2 Kmodulo norms from K.w/. 18 But every a 2 K is a norm from K.w/ in this case,

18K.w/ is a commutative separable algebra over K with an automorphism group G of order 2 (the groupinterchanging the two copies Ke1 and Ke2 of K). Thus K.w/ is a quadratic “Galois algebra”. The theory ofcrossed products of Galois algebras can be developed in complete analogy to the theory of crossed products forGalois field extensions. The first who had done this explicitly seems to be Teichmüller in his paper [Tei36b]. Histerminology was “Normalring” for what today is called “Galois algebra”. – Quite generally, for cyclic algebraswe refer the reader to [LR03] and to [Lor08a].


and therefore C.V / splits, i.e., it is a full matrix algebra over K. The formula (7.14)is still valid. In this way the case ab 2 }.K/ appears quite analogous to the caseab … }.K/.

But there is one essential difference. If ab … }.K/ then the quadratic space V isanisotropic, i.e., q.x/ D 0 only for x D 0. This we see from (7.14) since K.w/ isa field and therefore the norm function N.z/ ¤ 0 if z ¤ 0. But if ab 2 }.K/ thenV is isotropic since N.e1/ D N.e2/ D 0. In this case it turns out that the quadraticform is equivalent to q.x1; x2/ D x1x2. The corresponding quadratic space is calledthe hyperbolic plane and denoted by H .

In any case, the residue class of ab modulo }.K/ is an invariant of V since it isdetermined by K.w/ D C0.V /. This residue class is by definition the Arf invariantof the binary space V : 19

Arf.V / W ab mod }.K/: (7.16)

The above discussion shows the validity of the following

Theorem. Let V D KuCKv be a binary regular quadratic space, sothat (7.10) holds. The Clifford algebra C.V / is a quaternion algebraover K. Together with the Arf invariant Arf.V / the Clifford algebraC.V / completely determines V up to isomorphism. V is isotropic if andonly if Arf.V / 0 mod }.K/, and then V is the hyperbolic planeH .

Remark. Central simple algebras in every characteristicp > 0with defining relations

up D a; }.w/ D c; uw D .w C 1/u: (7.17)

had been systematically studied earlier, in particular by Teichmüller in his paper[Tei36a]. Such an algebra is called p-algebra. Teichmüller denotes it by .a; c �. 20

In the case p D 2 we obtain a quaternion algebra A. In view of the aboveconsiderations we see that A D .a; c � is the Clifford algebra C.V / of the binaryquadratic space V D Ku C Kv with (7.10) for b D a�1c. Whereas the relations(7.11) represent the description of A as the Clifford algebra of the given quadraticspace V , the relations (7.17) put into evidence the description of A as a crossedproduct of some separable quadratic extension K.w/ of K with }.w/ D c. Theconnection between the two is given by the fact that c is the Arf invariant of the spaceV , while the quadratic form of V is given by the norm form of K.w/ scaled by a;see (7.14).

The paper of Teichmüller mentioned above appeared in 1936, one year beforeArf came to Göttingen. In this paper Teichmüller studies, among other things, the

19Arf in his paper writes .V /.20Apparently this notation had been chosen to signalize the fact that the symbol .a; c � is not symmetric in a

and c. Compare it with the notation .a; b/ for a quaternion algebra in characteristic ¤ 2, given by the definingrelations (7.19) below. That symbol is symmetric in the sense that .a; b/ D .b; a/.


conditions for two such p-algebras .a; c � and .a; c 0 � to be isomorphic. If p D 2 thenthis result has some bearing on Arf’s investigations. It would have been desirablethat Arf cites Teichmüller’s paper and points out the connection between his andTeichmüller’s investigation. However Arf did not do this. Why not? Did he not knowTeichmüller’s paper?

Teichmüller had been a very active member of the Göttingen Arbeitsgemeinschaftbut he had left for Berlin in early 1937. Hence Arf had probably not met Teichmüller.But certainly Teichmüller’s results were known and valued in Göttingen and Arfmust have heard about it. I find an explanation for Arf’s silence about Teichmüller’swork in a certain character trait of Arf which is mentioned in the biography of Sertöz[Ser08]: 21

Arf was in the habit of encouraging young mathematicians to discovermathematics by themselves rather than to learn it from others. To supporthis cause he would tell how in his university years, i.e., his École Normaleyears in Paris, he would never attend classes … but proceed to developthat theory himself.

It seems that during his stay in Göttingen Arf had proceeded similarly, for Sertözreports in his biography:

Years later in Silivri, Turkey, Hasse would recall that after taking hisproblem22 Arf had disappeared from the scene for a few months only tocome back with the solution.

This suggests to me that when Arf in 1939 was back in Istanbul and worked onquadratic forms then again he had proceeded similarly, i.e., discovering the solutionof his problem by himself and not consulting other people or papers. In fact, in hispaper [Arf41] Arf cites only one paper explicitly, namely Witt’s on quadratic forms[Wit37a].

The above theorem holds in characteristic 2. Let us briefly compare it with thesimilar situation in Witt’s paper for characteristic ¤ 2: In this case a binary quadraticspace is of the form V D hu; vi with mutually orthogonal vectors u and v and insteadof (7.10), (7.11) we have

q.u/ D a; q.v/ D b; ˇ.u; v/ D 0: (7.18)

The Clifford algebra C.V / is now given by the defining relations

u2 D a; v2 D b; uv D �vu: (7.19)

21The story has been confirmed to me by several Turkish colleagues who had known Cahit Arf personally.22Namely the problem for his Ph.D. thesis.


Again, this is a quaternion algebra. In the theory of algebras it is often denoted by.a; b/. The even subalgebra C0.V / is of dimension 2 and has the basis 1; uv but thistime with the relation:

.uv/2 D �uv.vu/ D �u2v2 D �ab D d:

where d is the discriminant of V . Thus

C0.V / D K.pd/:

If d … K�2 then23 this is a quadratic field extension whose non-trivial automorphismis given by transformation with u. And again, we conclude that C.V / is a crossedproduct of K.

pd/ which splits if and only if a is a norm from L.

If d 2 K�2 then K.uv/ is not a field but the direct product of two copies of K.In this case and only in this case the quadratic space is isotropic, and it turns outthat in this case the corrsponding quadratic form is equivalent to q.x; y/ D xy, thehyperbolic plane.

So we see that for binary quadratic spaces Arf’s situation in characteristic 2 isquite similar to Witt’s situation in characteristic ¤ 2, the only difference being quitenatural, namely that the quadratic splitting field of C.V / is generated by

p�ab inthe case of characteristic ¤ 2, whereas in characteristic 2 it is generated by a root ofthe Artin–Schreier equation }.x/ D ab. And we see already here in the binary case:

In characteristic 2 the Arf invariant Arf.V / 2 K=}.K/ is the analogueof the discriminant d.V / 2 K�=K�2 in characteristic ¤ 2.

This was the guiding idea of Arf when he wrote his paper.

7.7 Higher dimensional quadratic spaces

Now let V be an arbitrary regular quadratic space over a field K of characteris-tic 2. From (7.5) we know that V decomposes into an orthogonal direct sum oftwo-dimensional spaces:

V D ?X

1�i�r

Vi where Vi D hui ; vi i (7.20)

andq.ui / D ai ; q.vi / D bi ; ˇ.ui ; vi / D 1 .1 � i � r/: (7.21)

23K� denotes the multiplicative group of the field K and K�2 is the group of squares.


Definition of Arf invariant.

Arf.V / X

1�i�r

Arf.Vi / mod }.K/: (7.22)

Recall that by definition Arf.Vi / aibi mod }.K/ so that this definition can alsobe written as:

Arf.V / X

1�i�r

q.ui /q.vi / mod }.K/: (7.23)

This formula is printed on the Turkish 10-Lira note where, however, the underlyingfield is restricted to be K D F2, the prime field in characteristic 2. In that case}.F2/ D 0 and hence the congruence sign in (7.23) can be replaced by equality.

If r > 1 then it is not clear a priori that Arf.V / is an invariant of V . For, thedefinition (7.23) depends on how V is decomposed into orthogonal subspaces Vi

in the form (7.20). One has to show that for every two such decompositions thecorresponding sums in (7.22) are in the same class modulo }.K/. Arf does it inhis paper but the proof requires some cumbersome computations. In later years Witt[Wit54] and Klingenberg [KW54] have given simplified descriptions of Arf.V / fromwhich one can see more directly its invariance. In the comments to Witt’s paper in[Wit98] the editor Ina Kersten reports:

It was Witt’s concern in the fifties to eliminate the assumption that thecharacteristic of the ground field is different from 2.

We interpret this such that Witt had carefully read Arf’s paper and tried not only tosimplify Arf’s proof but also to build a unified theory of quadratic forms, independentof the characteristic. In particular Kersten mentions Witt’s cancellation theorem (seeSection 7.8 below) and his attempts to investigate in detail the geometric situationwhich guarantees its validity.

Today we would verify the invariance of Arf.V / by investigating in more detailthe structure of the Clifford algebra C.V /. We have already said in Section 7.5that C.V / contains a subalgebra C0.V / which is canonically defined by V , namely:C0.V / is generated by the products with an even number k of factors in V . And inSection 7.6 we have seen that in the binary case, C0.Vi / D K.wi / is a quadraticextension defined by the relationw2

i Cwi D aibi which shows, using Artin–Schreiertheory, that the class Arf.Vi / of aibi is an invariant of C0.Vi /, hence of Vi . But Arfdid not consider the subalgebra C0.V /, probably he was not aware at that time thatC0.V / was canonically defined by the quadratic space V . Therefore he had to usesomewhat cumbersome explicit computations.

But using the invariance of C0.V /, the following statement immediately showsthat Arf.V / is an invariant:24

24I have found it in the book by Knus [Knu80].


Proposition. Let V be a regular quadratic space, represented as anorthogonal sum of two-dimensional spaces as in (7.20), (7.21). Foreach i let C0.Vi / D K.wi / with }.wi / D aibi . Put w D P

i wi ,so that }.w/ D P

i aibi . Then the quadratic extension K.w/ equalsthe center of C0.V /, and hence by Artin–Schreier theory the class ofP

i aibi is an invariant of V .25

The essential part of the proof consists in verifying w to commute with everyelement in C0.V /. I recommend to verify this for r D 2 (and then use induction).One has to use that

C0.V / D C0.V1/˝ C0.V2/C C1.V1/˝ C1.V2/

D K.w1; w2/C V1 ˝ V2

where C1.Vi / denotes the K-space generated by all products of an odd number ofelements inVi , henceC1.Vi / D Vi : Show thatw D w1Cw2 commutes withw1, withw2 and with every product x1x2 with xi 2 Vi . (Use the fact thatwixi D xiwi Cxi ).

Let us mention that in Witt’s situation of characteristic ¤ 2 there arises a problemwith the Clifford algebra C.V /. For, in general this is not a central simple algebraand it is not a product of quaternion algebras. For this reason in characteristic ¤ 2

Witt replaced the Clifford algebra C.V / by another algebra S.V / which Witt hascalled Hasse algebra; this is defined as follows: First recall the notation .a; b/ forthe quaternion algebra defined by the relations (7.19). Now consider the coefficientsai appearing in the diagonal form (7.3) and put di D a1a2 � � � ai . Then the Hassealgebra is defined as the n-fold tensor product

S.V / D ˝Y

1�i�n

.di ; ai / � ˝Y

1�i�j �n

.ai ; aj /:26 (7.24)

This is a central simpleK-algebra and plays a role in Witt’s theory of quadratic formsin characteristic ¤ 2, analoguous to the Clifford algebra in characteristic 2. But itsdefinition (7.24) depends on the coefficients ai in the diagonal form (7.3). In orderto show that it is an invariant, it is necessary to study the transformation from onediagonal form to an equivalent one. Witt’s computations for this are similar to Arf’scomputations for the invariance of Arf.V / in characteristic 2. It seems to me thatArf had modelled his invariance proof for Arf.V / after Witt’s invariance proof forS.V /. 27

25If }.w/ 0 mod }.K/ then K.w/ is not a field but the direct sum of two fields isomorphic to K. Wehave discussed this situation already in the case of two-dimensional quadratic spaces.

26Quite generally we write A B if A; B are central simple K-algebras which determine the same elementin the Brauer group Br.K/.

27We follow a suggestion of the referee and remark that the Hasse algebra S.V / in characteristic ¤ 2 is notan invariant of the class of V in the Witt ring WQ.K/ – contrary to the situation in characteristic 2 with theClifford algebra C.V / (see the next section).


7.8 Witt equivalence

For any type of mathematical structures, the quest for invariants is motivated by thehope to be able to characterize the structures by their invariants (up to isomorphisms),and thus to obtain a classification of the structures under investigation. Here we areconcerned with quadratic spaces V in characteristic 2 and in particular with regularspaces. We now know three invariants:

1. the dimension dim.V /,2. the Clifford algebra C.V / in the Brauer group Br.K/,3. the Arf invariant Arf.V / in the additive group K=}.K/.

For arbitrary fields we cannot expect that these three invariants characterize V upto isomorphisms. But Arf wished to show that for special fields K this is indeedpossible. Although, as we shall explain, his main result cannot be upheld in its fullgenerality, it turns out that the theorem is valid, e.g., over global and local fieldsK in characteristic 2. In order to approach this problem, Arf follows Witt who haddiscovered the “Witt ring” by introducing a certain equivalence relation.

Recall that a quadratic space V is called isotropic if there exists a non-zero vectorx 2 V with q.x/ D 0. The prototype of an isotropic regular space is the hyperbolicplane H already introduced in Section 7.6. The Arf invariant of H is Arf.H/ 0 mod }.K/, and the Clifford algebra is C.H/ � 1, which means that C.H/ splits.Arf proves the following

Kernel Theorem. (i) If the regular quadratic space V is isotropic thenV D H ? V 0 where V 0 is uniquely determined by V (up to isomor-phisms).

(ii) Consequently, every regular quadratic space V can be decomposedinto an orthogonal sum of a number of spaces isomorphic to H and aspace V � which is anisotropic, and V � is uniquely determined by V (upto isomorphisms).

The space V � is called the anisotropic kernel of V . Its quadratic form is called thekernel form 28 of V .

As a consequence of this result Arf proves the general

Cancellation Theorem. Suppose the quadratic space W is regular. Ifthere exist quadratic spaces V1; V2 such that W ? V1 Š W ? V2 thenV1 Š V2.

In characteristic ¤ 2 this famous cancellation theorem was contained in Witt’s paper.Arf has observed that it holds also in characteristic 2, but only if W is assumed to beregular.29

28In German: Grundform.


So Arf had obtained a new invariant of V , its anisotropic kernel V �. The originalspace V is obtained from V � by adding an orthogonal sum of a number of hyperbolicplanes, as many as the dimension of V requires. We note that

C.V / � C.V �/ and Arf.V / Arf.V �/ mod }.K/: (7.25)

since C.H/ � 1 and Arf.H/ 0 mod }.K/. We conclude:

In order to classify the quadratic spaces it is sufficient to classify theanisotropic spaces.

It is useful to work with the following

Definition ofWitt equivalence. Two regular quadratic spacesV ,W (orquadratic forms) areWitt equivalent if they have isomorphic anisotropickernels. (Notation: V � W .)

This is indeed an equivalence relation. It blends with the orthogonal sum, i.e., ifV1 � W1 and V2 � W2 then V1 ? V2 � W1 ? W2. 30 The Witt classes of regularquadratic spaces with the operation ? form a group which we denote by WQ.K/.We have V ? V � 0, i.e., the elements of this group are of order 2.

7.9 Arf’s Theorems

Now we are able to state the main result of Arf’s paper. For an algebraic functionfield or power series field K over a perfect base field of characteristic 2 he wishedto prove that the above three invariants completely characterize the regular quadraticspaces. The relevant property of these fields was, in his opinion, the following whichconcerns the Brauer group of K:

(Q): The quaternion algebras over K form a group within the Brauergroup Br.K/. In other words: If A and B are any quaternion algebrasover K then A˝ B � C where C is a quaternion algebra again.

It is not difficult to show that in characteristic 2 function fields and power seriesfields over a perfect base field have the property (Q). But it seems that Hasse had notseen it immediately and so he asked Arf about it, who replied in a letter of March 29,1940:

29On the other hand, there is no such restriction necessary for V1; V2. Consequently the definition belowof Witt equivalence applies to arbitrary quadratic spaces. But for simplicity we will restrict our discussion toregular spaces.

30If V and W are semi-regular but not regular then V ? W need not be semi-regular. Arf considers also thissituation but then the relation V ? V 0 does not hold generally.


Wenn A und B normale einfache Algebren vom Grade 2 sind, so istA˝B höchstens vom Index 4. Da aber ŒK

12 W K� D 2 so enthalten A

und B Teilkörper die zu K12 isomorph sind. A und B enthalten also

Elemente u; v mit u2 D v2 2 K die nicht zu K gehören. Es gilt daher

.u � v/2 D 0 ohne, dass u � v D 0 gilt.

A˝ B enthält also ein nilpotentes Element. Der Index von A˝ B istdaher höchstens 2.

If A and B are central31 simple algebras of degree 2 then the index ofA ˝ B is at most 4. But since ŒK

12 W K� D 2, both A and B contain

subfields which are isomorphic toK12 . HenceA andB contain elements

u and v respectively with u2 D v2 2 K, and u, v do not belong to K.Hence we have

.u � v/2 D 0 but not u � v D 0:

ThusA˝B contains a nilpotent element. Therefore the index ofA˝B

is at most 2.

This settled Hasse’s question but at the same time it showed that property (Q)holds for all fields with ŒK

12 W K� D 2.

Arf stated his main results in the form of two theorems. As indicated earlier, thereis an error in the proof of his first theorem and in fact there do exist counterexamples.Hence his first theorem has to be corrected. Nevertheless let us first state it as itappears in Arf’s paper.

Arf’s Theorem 1 (to be corrected).32 Assume that the field K of char-acteristic 2 satisfies property (Q). Then any regular quadratic space Vof dimension > 4 is isotropic. Consequently, its anisotropic kernel V �is of dimension � 4.33

Arf’s Theorem 2. Assume that the field K of characteristic 2 hasthe property that every regular quadratic space of dimension > 4 isisotropic. Then every regular quadratic space overK is uniquely deter-mined, up to isomorphism, by its dimension, its Clifford algebra and itsArf invariant.

31Arf used the terminology “normal” but nowadays it is usually said “central” to indicate that the center of thealgebra equals the base field. – The K-dimension of a central simple K-algebra A is a square n2. The numbern is called the “degree” of A. The “index” of A is defined to be the degree of the division algebra D A.

32In Arf’s paper [Arf41] this is Theorem 11, and our next Theorem 2 is numbered as Theorem 12 there.33Arf also considered quadratic spaces V which are semi-regular but not regular. For those he claimed that

the regular part of V � is of dimension � 2.


Certainly, Arf regarded his second theorem as the highlight of his paper. He hadbeen able to accomplish his aim, namely to characterize quadratic forms by theirinvariants. His first theorem was to give a sufficient criterion, in terms of the Brauergroup, for the field K which implies the characterization.

It can be easily verified that the condition (Q) is necessary for the validity of theassertion in Arf’s first theorem. But Arf was wrong to believe that it is also sufficient.In order to find the correct condition, necessary and sufficient, we first remark thatcondition (Q) is well known to be equivalent to the following condition:

(S)Any two quaternion algebrasA;B overK admit a commonquadraticsplitting field.

If two quaternion algebras have a common quadratic splitting field then they arecalled “linked”. If every two quaternion algebras overK are linked, i.e., ifK satisfiescondition (S), then K is called “linked”.

Every non-split quaternion algebra in characteristic 2 has two kinds of quadraticsplitting fields: separable and inseparable ones. If the quaternion algebrasA,B havea common inseparable quadratic splitting field then there is also a common separablequadratic splitting field. This seems to have first been observed by Draxl [Dra75]. Ashort and easy proof can be found in Lam’s paper [Lam02].34 See also Section 7.13.4below.

But now comes the surprise: the converse does not hold. If A, B have a com-mon separable quadratic splitting field then they do not necessarily have a commoninseparable quadratic splitting field. This has been observed by [Lam02].35 In viewof this the following condition appears stronger than (S):

(Sins) Any two quaternion algebrasA;B overK admit a common insep-arable quadratic splitting field.

Fields K with this property may be called “inseparably linked”. R.Aravire andB. Jacob [AJ95, AJ96] have shown that the iterated power series field F2..X//..Y //

is linked but not inseparably linked. We conclude that condition .Sins/ is properlystronger than (S), hence also properly stronger than Arf’s condition (Q) which isequivalent to (S).

It turns out that the proper correction of Arf’s first theorem consists of replacinghis condition (Q) by the stronger condition .Sins/. This has been shown by Baeza[Bae82]:

Baeza’s Theorem. (i) If K satisfies condition .Sins/ then every regularquadratic form over K of dimension > 4 is isotropic.

34Perhaps it is not without interest to note that the formulas for quaternion algebras which have been used in[Lam02] are special cases for p D 2 of formulas which have been stated 1936 by Teichmüller for p-algebras incharacteristic p [Tei36a].

35Lam cites [Tit93] and [Knu93] but his example is simpler and easier to verify.


(ii) Conversely, if every regular quadratic form overK of dimension> 4is isotropic then .Sins/ holds.

Quite generally, the so-called u-invariant u.K/ of a field K is defined to bethe smallest number u such that every regular quadratic form of dimension > u isisotropic. Thus Baeza’s theorem can be formulated as follows:

If K is inseparably linked then u.K/ � 4, and conversely.

If we observe that Arf’s first (incorrect) theorem can be formulated as: If K islinked then u.K/ � 4, then we see that Arf’s essential difference to Baeza’s theoremconsists in the absence of the inseparability condition. Apparently Arf was not awareof the fact that there is a difference of the linkage behavior of quaternions accordingto separability or inseparability. In fact, this question was first raised in 1974 only,by Draxl [Dra75].

If K is a function field of one variable over a perfect field of constants then thereis only one inseparable quadratic field over K, namely K

12 . Hence K is inseparably

linked and Baeza’s theorem is applicable. In fact, in this case this is almost trivial;see our appendix.

The classical fact that ŒK12 W K� D 2 for a function field K over a perfact base

field, was also observed by Albert in his paper [Alb38b] which we have cited abovealready. On this basis Albert had already proved that quadratic forms of 5 variablesover such function fields are isotropic, i.e., that u.K/ � 4.

Apparently Arf did not know Albert’s paper. When O. F. G. Schilling reviewedArf’s paper in the “Mathematical Reviews” he wrote: “The author is unaware of thework of A.A.Albert”. We observe that Schilling did say this as a statement, not as aguess. Schilling had been a student of Emmy Noether in Göttingen and after Noether’semigration got his Ph.D. with Helmut Hasse in Marburg. Later he went to the USA.36

At the time when he wrote this review he held a position at the University of Chicagowith Albert. He had kept contact to Hasse by mail, and on these occasions he hadasked for information about the results in Hasse’s Göttingen mathematical circle. Itseems likely that he had been informed by Hasse or by someone else from GöttingenaboutArf and his results; this enabled him to state thatArf “was not aware” ofAlbert’swork, and that he did not add “apparently” or something like this. Certainly Schillinghimself knew Albert’s papers.

Arf’s (erroneous) proof of theorem 1 is not easy or straightforward but it is wellarranged. It seems to me that Arf’s style in his paper was much influenced by thesuggestions and the advice of his academic teacher Hasse. For, I have found in Arf’spaper a footnote which Hasse, being the editor of Crelle’s Journal, had placed at theend of the introduction:

36He first stayed at the Institute for Advanced Study in Princeton where he had been accepted on the recom-mendation of Hasse who had written a letter to Hermann Weyl.


Anmerkung des Herausgebers: Im Einverständnis mit dem Verfasserhabe ich dessen ursprüngliches Ms. überarbeitet.

Note by the editor: With the consent of the author I have revised hisoriginal manuscript.

We see that Hasse did withArf’s manuscript what he always did as an editor of Crelle’sJournal, namely checking manuscripts carefully. As Rohrbach reports in [Roh98]:

With his [Hasse’s] characteristic conscientiousness, he meticulouslyread and checked the manuscripts word by word and formula by for-mula. Thus he very often was able to give all kinds of suggestions to theauthors, concerning contents as well as form …

So he did with Arf’s paper. In the Hasse–Arf correspondence we read several timesthat Arf responds to changes suggested by Hasse, both approvingly and critically.Finally on February 8, 1941 Arf returned the final version to Hasse and wrote:

Mit gleicher Post schicke ich Ihnen die Korrekturbogen und das Manu-skript der Arbeit über quadratische Formen zurück. Die Änderungen andrei Stellen die Sie vorgenommen haben scheinen mir unrichtig. MeineGründe habe ich am Rand des Manuskripts geschrieben.

At the same time I am returning the galley proofs and the manuscripton quadratic forms. At three instances your proposed changes seemnot to be correct. I have explained my reasons at the margin of themanuscript.37

The paper appeared in the same year 1941.It seems curious that Hasse had not detected Arf’s error although he was quite

interested in the subject and had closely examined Arf’s paper. This is even morecurious since the error is of the same kind which many years ago, in 1927, EmmyNoether had committed in a similar situation and there resulted a close correspondencebetween Hasse and Emmy Noether about it. This correspondence finally led to theirrenowned theorem about cyclicity of algebras over number fields. It appears that in1940 Hasse had forgotten that incident.

The situation back in 1927 had been as follows:38 Emmy Noether, in a letter toRichard Brauer of March 28, 1927, wrote to him that every minimal splitting field of adivision algebra can be embedded into the algebra. Brauer knew that this was not thecase and provided her with a counterexample. But this example seemed unnecessarilycomplicated to Emmy Noether; so she wrote to Hasse, in a postcard of October 4, 1927asking whether he could construct easy counterexamples for quaternions. Hasse didso: He constructed fields of arbitrary high degree (over Q) which were splitting fields

37The manuscript is not preserved. I do not know which changes Hasse had proposed.38I have told this story in detail in [Roq05].


of the classical quaternions but no proper subfield had this splitting property. ThusNoether as well as Hasse learned that one has to distinguish between minimal splittingfields and splitting fields of minimal degree; the latter indeed can be embedded intothe algebra.

Now, in Arf’s proof in the year 1940, a situation ocurred which was quite similarto the earlier one in 1927. On page 164, in the second paragraph of the proof of his“Satz 11”, Arf considered two quaternion algebras A1; A2 (i.e., Clifford algebras ofbinary quadratic spaces V1; V2). He took separable quadratic extensionsK1; K2 ofKwhich were splitting fields ofA1 andA2 respectively, and he assumedK1 ¤ K2. ThenA1 ˝A2 is split by the field compositumK1K2 which has degree 4 overK. In viewof his hypothesis (Q) Arf knew that A1 ˝ A2 is similar to some quaternion algebra,hence it admits a splitting field of degree 2. And now he argued that “necessarily”such splitting field can be found within K1K2. But this is not necessarily the case.In other words: K1K2 may be a minimal splitting field of A1 ˝A2 although it is nota splitting field of minimal degree.

Although Hasse’s example of 1927 had referred to quaternions over Q we havehere in characteristic 2 a similar situation. Why had Hasse not seen this error? Wewill never know. We know that in 1940 it was wartime and Hasse had been draftedto the Navy. He worked at a Navy research institute in Berlin and could attend to hisactivities as an editor of Crelle’s Journal in the evenings and on weekends only. Soit seems that he did not check Arf’s paper as thoroughly as he was used to in earliertimes with other papers for Crelle’s Journal.

7.10 An assessment of Arf’s paper

Arf’s paper [Arf41] is the first in which quadratic forms over arbitrary fields of char-acteristic 2 are systematically studied. It is true that there was already some workbefore Arf which was concerned with quadratic forms in characteristic 2, e.g., Dick-son in [Dic01] and Albert [Alb38b] – but these discussed only special base fields:finite fields and function fields of one variable, respectively. Due to Arf’s generalAnsatz he has opened the door to an extensive expansion of the theory of quadraticforms, not only over fields but also over arbitrary (commutative) rings.

Arf used the structural language, “modern” at his time, which had been introducedby Witt into the theory of quadratic forms. Thus he spoke of quadratic “spaces”instead of quadratic “forms”. Arf was able to extend a good part of Witt’s seminalresults in [Wit37a] to the case of characteristic 2. He showed the possibility of quasi-diagonalization, he extended Witt’s important cancellation theorem (Kürzungssatz) tocharacteristic 2, he investigated the role of Clifford algebras for quadratic forms, andhe defined his “Arf invariant” in characteristic 2 as a substitute for the discriminantin characteristic 0.


Arf’s theorems were meant to find conditions for fields K of characteristic 2which guarantee that all regular quadratic spaces over K are characterized (up toisomorphism) by their three invariants:

Dimension, Clifford algebra, Arf invariant.

His condition of linkage concerns the structure of the group of quaternion algebrasinside the Brauer group overK. Although the proof of his first theorem contained anerror and the theorem had to be modified, his paper still keeps its importance.

The error occurred due to the pathological and unforeseen behavior of quaternionalgebras in characteristic 2, which nobody of the time was aware of. In fact, it tookmany years until this was discovered and cleared up, and thus Arf’s first theoremcould be corrected [Bae82].

Due to Arf’s number theoretical background his main interest was directed toglobal and local fields K. In a letter to Hasse dated March 29, 1940 (from which wehave already cited on page 208) he had explained that any two quaternion algebrasover K have a common inseparable quadratic splitting field, namely K

12 . From this

he concluded that K is linked; this is true but it does not lead to the conclusion ofArf’s first theorem, namely u.K/ � 4. In fact, what he did show in his letter is thatK is inseparably linked and this, due to Baeza’a theorem, is sufficient for u.K/ � 4.

It seems not impossible thatArf’s first version of proof worked for local and globalfields of characteristic 2, and that he used correctly this inseparable linkage which hehad shown to Hasse. In any case, as will be put into evidence in our appendix, Arfhad all the ingredients of such proof at his disposal.

But since Arf did not realize the difference between linked and separably linkedquaternions, in his attempt to generalize his argument, he started with the linkagecondition (S) (or rather its equivalent condition (Q)), and since he could not provethat this implies inseparable linkage (which we know today is not true), he tried to useseparable quadratic splitting fields. And so, since he was convinced of his theoremdue to his experience with global fields, he stumbled into his error. And even Hasse,Witt, Albert and O. F. G. Scilling (among others) did not detect his error.

There are many examples in the history of mathematics showing that people,even respected and competent mathematicians, who are convinced of the validity ofa theorem, are apt to accept any decent looking proof even at the cost of overlookingsome little detail which then may necessitate a correction of the theorem.39

39One of those examples is Grunwald’s theorem in class field theory (1933), which was accepted by Artin,R. Brauer, Hasse and Albert (among others) until Wang presented a counterexample (1948). See, e.g., section 5of [Roq05].


7.11 Perfect base fields

If K is perfect of characteristic 2 then there exists only one quaternion algebra overK (up to isomorphisms), namely the split one. The linkage conditions (S) and also.Sins/ are trivially satisfied. Therefore, in the characterization of regular quadraticspaces the Clifford algebra can be omitted. Hence for a perfect base field, everyregular quadratic space V is characterized by its dimension and its Arf invariant.

But this result does not need Arf’s two theorems for its proof. For perfect basefields it follows almost immediately from the kernel theorem on page 206 that thekernel V � is of dimension � 2. If w is the Arf invariant of V (and V �) then thequadratic form of V � is given by the norm formula (7.14) on page 200, where we canput a D 1 since K is perfect. The original space V is obtained from V � by adding anumber of copies of the hyperbolic spaceH , as many as is required by the dimensionof V .

In the special case when K is finite then }.K/ is an additive subgroup of index2 in K and hence there is essentially one anisotropic quadratic space. Its quadraticform is q.x; y/ D x2 C xy C by2 where b … }.K/. If K D F2 then b D 1.

In the literature “Arf’s Theorem” is often confined to this case. For instance, inthe “Wikipedia”40 we read the following:

In mathematics, the Arf invariant of a nonsingular quadratic form overthe 2-element field F2 is the element of F2 which occurs most oftenamong the values of the form. Two nonsingular quadratic forms overF2 are isomorphic if and only if they have the same Arf invariant. Theinvariant was essentially known to Dickson (1901) and rediscovered byCahit Arf (1941).41

Certainly, this is all true. But does it give an idea about the main discovery ofArf ? In Arf’s paper the field F2 is not mentioned at all. In a small remark, coveringfour lines only, Arf mentions how his theory applies easily and almost trivially inthe case of perfect fields of characteristic 2. The main motivation and the mainresults of the theory of Arf invariants are concerned with fields which are not perfect.Arf studied the role of central simple algebras in the theory of quadratic forms incharacteristic 2. This aspect was not even scratched in that article of Wikipedia.Moreover, the definition of Arf invariant as given in that article is valid only for thebase F2 and does not apply to other fields of characteristic 2. 42

The Wikipedia article seems to be written in view of the application ofArf’s theoryin topology. For, several of those applications are mentioned in the article. And indeedin topology one has to compute cohomology and other functors with coefficients

40This refers to a former English version, August 20, 2009. A similar text, also restricted to the base field F2,appears in the “Encyclopedia of Mathematics”.

41It seems that the author tacitly assumes that both these quadratic forms have the same number of variables.42It is a good exercise to identify the defintion in that article with Arf’s definition when the base field is F2.


modulo 2 which means that the base field is F2. An overview of the application ofArf invariants in topology is given by Turgut Önder in the appendix of the CollectedPapers of Cahit Arf [Arf90].43 But, as said above, this is not representative of Arf’swork which is meant to exhibit the role of central simple algebras in the theory ofquadratic forms in characteristic 2.

By the way, in the 1901 book of Dickson on linear groups [Dic01] which ismentioned in the Wikipedia article, also the case of an arbitrary finite base field ofcharacteristic 2 is treated, not only F2. The fact thatArf did not cite this book may haveone of two reasons: either he knew Dickson’s book and found it is of no relevancefor his investigation (which would be understandable), or he did not know it (whichis more probable in view of his particular character trait which we have mentioned inSection 7.6).

In any case, the statement that Arf has “rediscovered” what Dickson had knownis misleading. Arf discussed a quite different theorem, and in a very special case thisimplies the statement of Dickson.

7.12 Epilog

After my conference talk I was asked aboutArf’s biography for the years after his paperon quadratic forms. I will not repeat here what is said in his biographies containedin [Arf90, Ser08]. Let me only mention that he became a prominent member ofthe Turkish scientific community (which is documented by the fact that his portraitdecorates an official banknote) – but he also was a dedicated teacher. Many youngermathematicians in Turkey had been introduced by him into mathematics, he hadencouraged them and showed them understandingly the way into our science. He iswidely remembered in the mathematical community of Turkey. Robert Langlands, inhis article about his impressions in Turkey, remembers warmly his discussions withArf [Lan04]. In particular Arf had directed Langlands’ attention to a paper by Hasseon the local decomposition of the "-factors; these factors appear in the functionalequation of Artin’s L-series. As Langlands says (English translation):

I had rapid advance in my research having read Hasse‘s paper…

and

…thanks to Cahit bey, I solved this problem during my stay in Ankaraand proved the existence of the local "-factor.

I was also asked to report more extensively on the correspondence between Arfand Hasse, in particular the letters after 1941. I plan to do this some time in the

43For the so-called Arf–Kervair invariant see, e.g., the article by Snaith in Morfismos Vol. 13 (2009), no. 2,1–53, see also arXiv:1001.4751v1 [math.AT].

http://arxiv.org/abs/1001.4751v1


future. Let me only mention that these letters, although they do not discuss anymore mathematics proper, show a growing friendship between the two. Hasse visitedTurkey several times between 1957 and 1975. The last two preserved letters, datedMarch 1975, concern the proposal to have an international colloquium on the structureof absolute Galois groups. This colloquium was planned by Arf jointly with M. Ikeda(who had earlier got a position in Turkey on the recommendation of Hasse). Thisconference took place in September 1975 in Silivri, a small village on the beach ofthe Marmara sea. I had the chance to participate in this conference and was ableto observe the close friendly relationship between the two mathematicians, Arf andHasse.

7.13 Appendix: Proofs

The aim of this section44 is to put into evidence that the proof of Baeza’s theorem,i.e., the correction of Arf’s first theorem, can be done solely with the arguments whichcan be found in Arf’s paper. Hence Arf could well have proved Baeza’s theorem, thusavoiding his error, if only he would have recognized the difference between linkageof separable and inseparable type.

We also add a simple proof of Arf’s second theorem, as well as of Draxl’s lemma.We believe that our proofs, based on Arf’s paper, are simpler than any of those

which can be found in the literature.

7.13.1 Baeza’s Theorem (i)

K denotes a field of characteristic 2. We use the following notation:Let V be a regular quadratic space of dimension 2 over K. There is a K-basis

u; v of V with

q.u/ D a; q.v/ D b; ˇ.u; v/ D 1 .a; b 2 K/: (7.26)

Here, q W V ! K denotes the quadratic form of V and ˇ is the corresponding bilinearform. Let A D C.V / be the Clifford algebra of V . This is a quaternion algebra overK with basis 1; u; v; w and the relations

u2 D a; v2 D b; uv C vu D 1; w D uv: (7.27)

We identify the quadratic space V with the subspace of A generated by u and v, andthen q.x/ D x2 for all x 2 V . If x2 … K2 then K.x/ is an inseparable quadraticsubfield of A.

Lemma 1. Let y 2 A. Then y2 2 K if and only if y 2 V CK.

44This section has been written jointly with Falko Lorenz.


IfA does not split then this lemma gives a complete description of the inseparablequadratic subfields L D K.y/ � A : then y D x C c with suitable x 2 V , c 2 K.

The statement of Lemma 1 can be found in Arf’s paper [Arf41] on page 161.Arf does not formulate the statement in the form of a lemma, he just performs thecomputations which we give in the proof below and uses them in his text.45

Proof of Lemma 1. We represent y 2 A in the form

y D x C z with x 2 V; z D c0 C c1w 2 K.w/; c0; c1 2 K; (7.28)

and compute

y2 D x2 C z2 C xz C zx

D x2 C z2 C c1.xw C wx/

D x2 C z2 C c1x (7.29)

where we have used that xw C wx D x for x 2 V , which is a consequence ofthe relations (7.27). Now, x2 D q.x/ 2 K, z2 2 K.w/ and c1x 2 V . SinceV \K.w/ D 0 we conclude that y2 2 K if and only if c1x D 0 and z2 2 K, hencec1 D 0 since K.w/jK is separable.

In the following we regardK as a 1-dimensional quadratic space with the quadraticform q.c/ D c2 for c 2 K. The sum V C K � A is direct and can be regarded asthe orthogonal sum V ? K of quadratic spaces.

Lemma 2. LetV; V 0 be regular quadratic spaces of dimension 2 and letA;A0 be theirClifford algebras. Assume that A and A0 do not split. Then A;A0 have a commoninseparable quadratic splitting field if and only if V ? V 0 ? K is isotropic.

Proof. LetK.y/ � A andK.y0/ � A0 be isomorphic inseparable quadratic subfieldsof A and A0 respectively. We choose the generators y; y0 in such a way that theycorrespond to each other in the isomorphism K.y/ Š K.y0/, so that y2 D y0 2. ByLemma 1 we have y D x C c with 0 ¤ x 2 V and c 2 K. Similarly y0 D x0 C c0with 0 ¤ x0 2 V 0 and c0 2 K. We conclude

x2 C c2 D x0 2 C c0 2:

Putting d D c C c0 we obtain

x2 C x0 2 C d2 D 0 (7.30)

which shows that the quadratic space V ? V 0 ? K is isotropic.

45Arf’s notation is different from our’s.


Conversely, assume V ? V 0 ? K is isotropic. There is a nontrivial relation ofthe form (7.30) with x 2 V , x0 2 V 0, d 2 K. It follows

x2 D .x0 C d/2: (7.31)

Let, say, x ¤ 0. Since A does not split we have x2 … K2. Hence K.x/ is aquadratic inseparable subfield of A. From (7.31) we conclude that K.x0 C c/ � A0is isomorphic to K.x/.

Proof of Baeza’s theorem, part (i) (see page 209). We assume that every two non-split quaternion algebras overK have a common inseparable quadratic splitting field.We claim that every regular quadratic space of dimension> 4 is isotropic. Write thisspace in the form

V ? V 0 ? W

where V and V 0 are of dimension 2 and W of dimension > 0.Let y 2 W and assume first that q.y/ D 1. Then the subspace Ky � W is

isomorphic to K as a quadratic space. Since the Clifford algebras C.V / and C.V 0/have a common inseparable quadratic splitting field (by assumption) we infer fromLemma 2 that V ? V 0 ? Ky is isotropic. Hence V ? V 0 ? W is isotropic too.

But W may not contain a vector y with q.y/ D 1. In this case we use themethod of scaling (see page 198). If 0 ¤ c 2 K then the scaled quadratic spaceV .c/ ? V 0 .c/ ? W .c/ is isotropic if and only if V ? V 0 ? W is isotropic. Nowchoose y 2 W with q.y/ ¤ 0 and take the scaling factor c D q.y/�1. Thenc � q.y/ D 1 which means that the scaled space .Ky/.c/ is isomorpic to K. Fromwhat has been shown above it follows that V .c/ ? V 0 .c/ ? W .c/ is isotropic, henceso is V ? V 0 ? W .

7.13.2 Arf’s second theorem

We assume that every regular quadratic space of dimension > 4 over K is isotropic.We claim that every regular quadratic space over K is uniquely determined (up toisomorphism) by its dimension, the Brauer class of its Clifford algebra and its Arfinvariant.

Proof. Recall that WQ.K/ denotes the (additive) group of Witt classes of regularquadratic spaces overK; see Section 7.8. The Witt class of the space V is representedby its anisotropic kernel V �. The whole space V arises from V � by adding a numberof hyperbolic planes H , as many as the dimension of V requires.

Hence our claim reduces to the claim that the Witt class of V , i.e., the anisotropicspace V �, is uniquely determined by its Brauer class of C.V / and the Arf invariantArf.V /.

We recall that

C.V �/ � C.V / and Arf.V �/ Arf.V / mod }.K/:


The Clifford algebra yields a homomorphism V 7! C.V / of WQ.K/ into the Brauergroup Br.K/. And theArf invariant yields a homomorphismV 7! Arf.V / of WQ.K/into the }-factor group K=}.K/. We have to show:

IfC.V / � 1 in Br.K/ and Arf.V / 0 mod }.K/ then V � 0 in WQ.K/, whichis to say that the anisotropic kernel V � of V vanishes.

We have dim V � � 4 by hypothesis. Since V � is regular we have dim V � D 0; 2

or 4. The case dim V � D 2 is not possible since Arf.V �/ 0 mod }.K/ impliesV � D H , the hyperbolic plane, hence V � would not be anisotropic.

Suppose dim V � D 4 and write V � D V1 ? V2 as the orthogonal sum of twobinary spaces. Since C.V �/ � 1we have C.V1/ � C.V2/. Since both algebras havethe same dimension it follows that they are isomorphic: C.V1/ D C.V2/. Also, sinceArf.V �/ 0 mod }.K/ we have Arf.V1/ Arf.V2/ mod }.K/. We have seen inSection 7.6 that a binary regular space is uniquely determined by its Clifford algebraand its Arf invariant. It follows V1 D V2 hence V � D V1 ? V1 � 0, so again V �would not be anisotropic.

7.13.3 Baeza’s theorem (ii)

The following lemma is the separable analogue to Lemma 1. The situation is thesame as in Lemma 1.

Lemma 3. Let y 2 A. Then }.y/ 2 K if and only if y 2 V CK C w.

If A does not split then this lemma gives a complete description of the separablequadratic subfields L D K.y/ � A : then y D x C c C w with suitable x 2 V ,c 2 K.

The statement of Lemma 3 can also be found in Arf’s paper [Arf41] on page 165.Arf does not formulate this statement in the form of a lemma, he just performs thecomputations which we give in the proof below and uses them in his text. 46

Proof of Lemma 3. We represent y 2 A in the form as before in (7.28) and compute:

}.y/ D y2 � y D x2 � x C z2 � z C c1x

D x2 C .c1 � 1/x C }.z/ (7.32)

where we have used that xz C zx D c1x. We conclude that }.y/ 2 K if and only ifc1 D 1 and so z D c0 C w, hence y 2 V CK C w.

Changing notation, we write c D c0 and hence y D x C c C w. Observing that}.c C w/ D }.c/C }.w/ we have shown that

}.y/ D }.x C c C w/ x2 C }.w/ mod }.K/: (7.33)

We will have occasion to use this formula later.46Our notation differs from Arf’s notation.


In the following lemma we consider K.w C w0/ � A˝ A0 as a quadratic spacewith respect to its norm function N W K.w C w0/ ! K. Explicitly we have

N.c0 C c1.w C w0// D c20 C c0c1 C c2

1}.w C w0/: (7.34)

The sum V C V 0 C K.w C w0/ � A ˝ A0 is direct and can be regarded as theorthogonal sum V ? V 0 ? K.w C w0/ of quadratic spaces.

Lemma 4. Let V , V 0 be regular quadratic spaces of dimension 2 and let A, A0 betheir Clifford algebras. Assume thatA andA0 do not split. ThenA,A0 have a commonseparable quadratic splitting field if and only if V ? V 0 ? K.w C w0/ is isotropic.

Proof. Let L � A and L0 � A0 be isomorphic separable quadratic subfields ofA and A0 respectively. We may write L D K.y/ and L0 D K.y0/ where y, y0correspond under the isomorphism L � L0 and hence }.y/ D }.y0/ 2 K. ByLemma 3 we have y D x C c C w and y0 D x0 C c0 C w0 with suitable x 2 V ,x0 2 V 0, c; c0 2 K. From (7.33) we obtain:

x2 C }.w/ x0 2 C }.w0/ mod }.K/;

x2 C x0 2 }.w/C }.w0/ mod }.K/; (7.35)

x2 C x0 2 D }.d/C }.w C w0/ with d 2 K: (7.36)

hencex2 C x0 2 C }.d C w C w0/ D 0:

Here, }.d C w C w0/ D N.d C w C w0/; see (7.34). It follows that the quadraticspace V ? V 0 ? K.w C w0/ is isotropic.

Conversely, assume that V ? V 0 ? K.wCw0/ is isotropic. We have to show thatthere exists a common separable quadratic splitting field of A and A0. There exists anontrivial relation of the form

x2 C x0 2 CN.z/ D 0 with x 2 V; x0 2 V 0; z 2 K.w C w0/:

We write z D c0 C c1.w C w0/ and use formula (7.34) for the norm. It follows

x2 C x0 2 C c20 C c0c1 C c2

1}.w C w0/ D 0 (7.37)

Suppose first that c1 ¤ 0. After dividing by c21 on both sides in (7.37) and

changing notation we may assume c1 D 1. Hence

x2 C x0 2 C }.c0 C w C w0/ D 0

which givesx2 C }.w/ x0 2 C }.w0/ mod }.K/


and using (7.33):}.x C w/ }.x0 C w0/ mod }.K/:

It now follows that the quadratic Artin–Schreier extensions K.x C w/ � A andK.x0 C w0/ � A0 are isomorphic.

If c1 D 0 then (7.37) shows that V ? V 0 ? K is isotropic. From Lemma 2we infer that A and A0 have a common inseparable splitting field. We have alreadymentioned Draxl’s lemma which says that then there is also a common separablesplitting field. For a simple proof of Draxl’s lemma see the next Section 7.13.4.

Proof of Baeza’s theorem part (ii) (see page 209). We assume that every regularquadratic space of dimension > 4 is isotropic. We claim that every two non-splitquaternions A, A0 over K have a common inseparable quadratic splitting field.

We write A D C.V / as the Clifford algebra of a 2-dimensional regular quadraticspace V as in (7.26), (7.27), and similarly A0 D C.V 0/. As above we considerthe separable quadratic extension K.w C w0/ � A ˝ A0 as a quadratic space withrespect to the norm. The 6-dimensional space V ? V 0 ? K.w C w0/ is isotropic(by assumption) and hence Lemma 4 shows that A and A0 have a common separablequadratic splitting field L. But we are looking for a common inseparable quadraticsplitting field; this will be established as follows.

The common separable quadratic splitting fieldL can be embedded intoA and intoA0; this yields isomorphic separable quadratic subfields inA and inA0. After changingnotation we now identify these two fields, so that A, A0 appear as crossed productsof the same separable quadratic field L. Writing L D K.w/ with }.w/ D c 2 K

we have A D .a; c � and A0 D .a0; c � with certain a; a0 2 K which represent thefactor systems of LjK defining A and A0 respectively. We refer to our discussionon page 201. As explained there, A, A0 appear now as the Clifford algebras of thequadratic spacesL.a/ andL.a0/ respectively, which are scaled quadratic spaces of thespace L with respect to the norm.

Now, by hypothesis the 6-dimensional space

L.a/ ? L.a0/ ? L

is isotropic. Thus there is a nontrivial relation of the form

aN.z/C a0N.z0/CN.y/ D 0 with z; z0; y 2 L: (7.38)

If y ¤ 0 then after dividing by N.y/ and changing notation we obtain

aN.z/C a0N.z0/C 1 D 0 with z; z0 2 K.w/.This shows that L.a/ ? L.a0/ ? K is isotropic. Applying Lemma 2 we see that thereexists a common inseparable quadratic splitting field of A and A0.

If y D 0 then from (7.38) we infer that L.a/ ? L.a0/ is isotropic, hence L.a/ ?L.a0/ ? K is isotropic too and again Lemma 2 applies.


7.13.4 Draxl’s Lemma

Lemma 5 (Draxl). Let A and A0 be two nonsplitting quaternion algebras over K.If A;A0 have a common inseparable quadratic splitting field then they also have acommon separable quadratic splitting field.

Proof. We represent A in the form

A D V CK.w/ with V D KuCKv, w D uv (7.39)

as explained in Section 7.6, and similarly A0. Let L be a common inseparablequadratic splitting field of A and A0. Embedding L � A we have L D K.x C c/

with x 2 V , c 2 K, according to Lemma 1 on page 216. We may assumethat v is linearly independent of x C c (otherwise interchange v and u). WritingQu D x C c, Qv D v, zw D Qu Qv we have another representation of the same kind as(7.39): zA D zV C K. zw/ : : : Changing notation, we omit the tilde and now have thesituation (7.39) with L D K.u/.

Similarly, we take a subfield L0 � A0 isomorphic to L and we adapt the repre-sentation of A0 such that

A0 D V 0 CK.w0/ with V 0 D Ku0 CKv0, w0 D u0v0

and L0 D K.u0/ where u and u0 corespond to each other under the isomorphismL � L0, hence

u2 D a D u0 2:

We have to find isomorphic separable quadratic subfields of K.y/ � A andK.y0/ � A0. To this end we put

y D cuC v C w and y0 D cu0 C w0

where the coefficient c 2 K will be determined below. Using (7.33) we compute(congruences are mod }.K/):

}.y/ .cuC v/2 C }.w/

c2aC b C c C }.w/ I}.y0/ c2aC }.w0/:

We see that

}.y/ }.y0/ ” c b C }.w/C }.w0/:

If c is chosen that way then the separable quadratic Artin–Schreier fields K.y/ � A

and K.y0/ � A0 are isomorphic.

Chapter 8

Hasse–Arf–Langlands

Introduction of Robert Langlands at the Arf lecture, November 11, 2004.

Middle East Technical University, Ankara.

In those years, each year there was scheduled a special colloquium lecture at the Middle EastTechnical University in Ankara, commemorating Cahit Arf who had served as a professor formany years at METU. In the year 2004 the invited speaker was Robert Langlands, and I hadbeen asked to chair his lecture. The following text was read to introduce the speaker. I tried topoint out a historic thread from the work of Helmut Hasse to Robert Langlands, via Cahit Arf.

It is my pleasure and a great honor to me to introduce Robert Langlands as thespeaker of the Arf Lecture of this year 2004, here at the Middle East TechnicalUniversity.

Professor Langlands is a permanent member of the Institute for Advanced Studyin Princeton, which is one of the most prestigious places for mathematical researchstudies. He has received a number of highly valued awards, among them the Cole Prizein Number Theory from the American Mathematical Society in 1982, the NationalAcademy of Sciences reward in Mathematics in 1988, and the Wolf prize from theWolf foundation in Israel 1996. He is elected fellow of the Royal Society of Canadaand the Royal Society of London, and he has received honorary doctorates from atleast 7 universities (to my knowledge).

In the laudation for the Wolf prize it was said that he received the prize for his

path-blazing work and extraordinary insights in the fields of numbertheory, automorphic forms and group representations.

Now, while I was preparing the text for this introduction it soon became clear tome that in this short time I would not be able to describe, not even approximately,Langlands’ rich work, its underlying ideas, its enormous impact on the present math-ematical research world wide, and its consequence for the future picture of Math-ematics. After all, our speaker today with his own words will be the best man forexplaining the main lines of his vision.

Instead, please allow me to add some comments of a more personal nature.Perhaps I should explain that my own mathematical interests at present are pre-

dominantly of historical nature. I am interested in particular in the transmission of

224 8 Hasse–Arf–Langlands

ideas from one mathematician to another, and from one generation to the next. Math-ematics does not develop just by itself, it is shaped by people. In the process ofhistory of science the communication between people plays a decisive role, and I findit fascinating to follow the lines of communication along which mathematical ideashave developed in the past.

Under this aspect I recently have found a very interesting statement by ProfessorLanglands himself. I have found it in an article in which he recollects his impressionsduring his first visit to Ankara in the academic year 1967–68. The article is writtenin Turkish language but a friend kindly translated it for me into English. From thisI learned that Langlands’ year at Ankara was quite decisive for his own research, inparticular through his contact with Cahit Arf. Langlands writes that Arf had pointedout to him a paper by Helmut Hasse who had proved the first results in the directionwhich he, Langlands, was exploring just then. Citing Langlands’ own words:

Thanks to Cahit bey, I solved this problem during my stay in Ankara andproved the existence of the local "-factor. Perhaps this theorem wouldnever have been proved if I had been somewhere else that year…

Indeed, I find this is a very interesting statement.As a young student, Cahit Arf had worked with Hasse in Göttingen to obtain his

doctorate. This was in the years 1937–38. In this formative period of his life, Arfbecame familiar with the mathematical ideas of that time in the circle around Hasse.Many of those ideas were directed towards the generalization of class field theory,from abelian to general Galois field extensions.

Perhaps I may be allowed here to use some technical mathematical expressionswhose precise meaning I cannot define here. I hope that in any case I am able to conveyto you that in the 1930s there were strong attempts concerned with a theory, calledclass field theory, which had been inherited from many generations of mathematiciansand now was aimed at generalization to a much wider scope than it was known at thetime. In fact, the topic of Arf’s doctoral thesis was a direct fall-out of those attempts.The main result of his thesis, nowadays known as the “Hasse–Arf Theorem” (moreprecisely: Arf’s part of the Hasse–Arf theorem), had been envisaged already in 1930by Artin in a letter to Hasse. Thus Arf’s thesis was fulfilling eight years later whathas been called “Artin’s dream”.

Helmut Hasse and Emil Artin belonged to the group of leading figures at thetime of the said activities; other names were Emmy Noether, Claude Chevalley andShokichi Iyanaga, André Weil. As I have mentioned, Cahit Arf was also working onthose projects.

However, despite strong and penetrating attempts from the side of those peopleand other mathematicians, the desired general results for Galois extensions could notbe reached in the 1930s, and class field theory remained to be essentially an affair ofabelian extensions only. In a letter from Artin to Hasse dated March 9, 1932 Artinsaid:

8 Hasse–Arf–Langlands 225

I have the impression that there has to be a completely new idea in orderto obtain the desired general results.

A similar mood can be read from Emmy Noether’s letters to Hasse, but she wassomewhat more specific already. On April 26, 1934 she wrote:

It seems to me that first we have to clear up the situation with respect tothe irreducible representations, which means the Galois modules, andat the same time to Artin’s conductors.

Even one week before Noether’s death in 1935 she wrote that she was still thinkinghard about this problem.

Certainly, Arf had become familiar with this situation when in 1937 he studiedwith Hasse; this can be concluded from his letters which are preserved in the library ofGöttingen University. Also,Arf knew that a certain problem connected with the theoryof Artin’s L-functions, namely the existence of the local "-factor, was considered tobe a step towards further development. But in the 1930s, this and the big problemof Galois class field theory remained unsolved, with little essential progress in thefollowing decades.

Finally, 30 years later in 1967 Langlands, after discussing with Arf and afterstudying Hasse’s work, succeeded to have the “completely new idea” which Artinhad wishfully foreseen in his letter to Hasse. And it was connected with Artin’sconductors and with irreducible Galois representations, like Noether had envisaged.This was a step forward in the development of what is now called the Langlandsprogram which implies a new form of class field theory valid for Galois extensions,and which Langlands had outlined in his famous letter to André Weil 1967.

H. Hasse C. Arf R. Langlands

Let me repeat: The historical moment when the first inspiration for the "-factorcame, was in 1967/68 and, according to Langlands himself, it happened here at theMiddle East Technical University, through the conveyance of Arf who had kept close

226 8 Hasse–Arf–Langlands

and friendly relationship to his former teacher Hasse all through his life time, and whothus was informed about Hasse’s ideas and work. In this way Langlands has becomepart of a long historical line of mathematical development, through Arf directly toArtin, Hasse and Noether, and from there on far back through Hecke, Takagi andHilbert to Abel.

Certainly this will not be the only historical line which can be found leading toLanglands’ widespread work. But today, at this occasion, this line through Arf seemsto be of particular significance.

On this background I feel that this year’s Arf lecture, here at the Middle EastTechnical University, will become something very special.

Professor Langlands, on behalf of the Mathematics Department at METU I havethe honor to invite you to start with your address.

Chapter 9

Ernst Steinitz and abstract field theory

In Memoriam Ernst Steinitz.

Journal für die reine und angewandte Mathematik (Crelle’s Journal) 658 (2010), 1–11.

Ernst Steinitz

In the year 1910, in volume 137 of Crelle’s Journal there appeared a paper with thetitle

Algebraische Theorie der Körper (Algebraic Theory of Fields).

The author was Ernst Steinitz. Let us use this occasion of a centenary to recall theimpact which Steinitz’s paper had upon the mathematicians of the time, and its rolein the development of today’s algebra. Bourbaki [Bou60] states that this paper hasbeen

228 9 Ernst Steinitz and abstract field theory

… un travail fondamental qui peut être considéré comme ayant donnénaissance à la conception actuelle de l’Algèbre.

… a fundamental work which may be considered as the origin of today’sconcept of algebra.

One of the first eager readers of Steinitz’s paper was Emmy Noether. At the timewhen the paper appeared she was still living in her hometown Erlangen, during whatmay be called the period of her apprenticeship, studying the highlights of contempo-rary mathematics of the time. Her guide and mentor in this period was Ernst Fischer.We can almost be sure that Steinitz’s paper was the object of extensive discussionsbetween Fischer and Emmy Noether.1

Steinitz’s ideas contributed essentially to the shaping of Emmy Noether’s conceptof mathematics and in particular of algebra. During the next decade every one ofNoether’s papers (except those on mathematical physics) contains a reference toSteinitz [Ste10].

Later in Göttingen, when she had started her own “completely original mathemat-ical path”2 she took Steinitz’s results as “well known” and she used them as the basisfor her work on fields and rings – in particular in her project to reformulate classicalalgebraic geometry in terms of abstract commutative algebra. She urged her studentsand her fellow mathematicians to study the classical papers which she consideredto be the roots of abstract algebra – among them was invariably the 1910 paper bySteinitz. Van der Waerden reports in [vdW75]:

When I came to Göttingen in 1924, a new world opened up before me. Ilearned from Emmy Noether that the tools by which my questions couldbe handled had already been developed by Dedekind and Weber, byHilbert, Lasker and Macaulay, by Steinitz and by Emmy Noether herself.She told me that I had to study the fundamental paper of E. Steinitz …

Van der Waerden’s textbook Moderne Algebra [vdW30], which was based onlectures by Emmy Noether and Artin, contains in his first volume a whole chapterabout Steinitz’s theory. Van der Waerden says in [vdW75]:

In earlier treatises, number fields, and fields of algebraic functions wereusually treated in separate chapters, and finite fields in still another chap-ter. The first to give a unified treatment, starting with an abstract def-inition of “field”, was E. Steinitz in his 1910 paper. In my Chapter 5,called “Körpertheorie”, I essentially followed Steinitz …

Van der Waerden’s Moderne Algebra was widely read and translated into manylanguages; in this way Steinitz’s ideas became known worldwide as part of the basics

1The mathematical letters between Fischer and Noether are preserved in the archive of the University ofErlangen. – Fischer’s name is still remembered today from the “Riesz–Fischer Theorem” in the theory of Hilbertspaces.

2Quoted from Alexandrov’s obituary for Emmy Noether [Ale83].


of contemporary algebra, and they found their way into the syllabus of beginnercourses. Almost all the notions and facts about fields which we teach our students insuch a course, are contained in Steinitz’s paper.

But what are those notions and facts? Let us point out first that the main point ofSteinitz’s paper was his abstract approach. As Purkert says in his essay on the genesisof abstract field theory [Pur73]:

Hier wurde erstmalig eine abstrakte algebraische Struktur auf der Grund-lage ihres Axiomensystems zum Gegenstand der Untersuchung gemacht.Dieses formalalgebraische Denken einerseits, die Verbindung mit derMengenlehre andererseits – das sind die Charakterzüge der modernenstrukturellen Algebra.

This was the first time that an abstract algebraic structure was studied onthe basis of its system of axioms. On the one hand the formal algebraicthinking, on the other hand the connection to set theory – these are thecharacteristics of the modern algebra of structures.

Hence, when we now describe the content of Steinitz’s paper we have to keepin mind that not only are those definitions and theorems important, but also the factthat they were obtained in the abstract axiomatic setting, notwithstanding the fact thatsome of them had already appeared in earlier treatises for fields of special kinds (fieldsof numbers, of functions, and finite fields). Steinitz’s paper was the first systematicinvestigation of the structure of abstract fields.3

In his preface Steinitz states that he wishes

eine Übersicht über alle möglichen Körpertypen zu gewinnen und ihreBeziehungen untereinander in ihren Grundzügen festzustellen.

to obtain a general overview of all possible types of fields and to deter-mine their relations with each other.

In any abstract field, Steinitz showed that there is a unique smallest subfield whichhe called the prime field; this is either infinite (and then isomorphic to the rationals) orits cardinality is a prime numberp (and then it is isomorphic to the integers modulop).Accordingly he defined the characteristic of a field to be either 0 or p respectively.Any integral domain determines a unique field of quotients; this is a frequently usedmethod to construct fields. For any field K and an irreducible polynomial f .X/ 2KŒX� there is an extension field L containing a root # of f .X/. If L is minimalwith this property then L is uniquely determined up to K-isomorphism. Steinitzdiscovered that in prime characteristic an irreducible polynomial may have multiple

3Actually, an earlier article by Heinrich Weber [Web93] also works in the framework of abstract field theory.Steinitz mentions that in his article. But he points out that Weber’s investigation is directed to Galois theory onlywhile his (Steinitz) goal is the systematic investigation of the structure of all fields. As said in [Kle99]: “WhileWeber defined fields abstractly, Steinitz studied them abstractly.”


roots, and so one has to distinguish between separable and inseparable algebraicextensions.4 Galois theory in this abstract setting holds for separable extensionsonly. Steinitz defined the notion of transcendence degree and he showed that everyfield can be obtained as an algebraic extension of a purely transcendental field, i.e.,a field of rational functions over the prime field. Finally, he constructed for everyfield an algebraic closure and showed that it is unique up to isomorphism – this isprobably Steinitz’s most important result. This theorem on the algebraic closure ofany field is sometimes considered to be the proper Fundamental Theorem of Algebra.

All the above has been included by van der Waerden in the first volume of histextbook Moderne Algebra. This appeared in 1930. The second volume, appearingone year later, contains the beginning of modern algebraic geometry in the frameworkof commutative algebra. In his Heidelberg lecture [vdW97] van der Waerden givesa lively account of how he discovered the algebraic definition of generic point anddimension of an algebraic variety – and in this connection he again refers to Steinitz’spaper. He says about it:

Die Wichtigkeit dieser Arbeit kann man garnicht überschätzen, das Er-scheinen dieser Arbeit war ein Wendepunkt in der Geschichte der Alge-bra des 20. Jahrhunderts.

One cannot overestimate the importance of this paper. The appearanceof this paper marks a turning point in the history of algebra of the 20thcentury.

Immediately after the publication of van der Waerden’s book it was rated, by thereferee in Jahrbuch für die Fortschritte der Mathematik, to be the

Standardwerk der modernen Algebra in der ganzen mathematischenWelt.

standard treatise of modern algebra in the whole mathematical world.

Van der Waerden’s was not the first textbook in which Steinitz’s theory was in-corporated. Two such textbooks had appeared earlier. One of them was Haupt’stwo-volume Algebra which appeared in 1929 [Hau29]. This book too had beenwritten under the influence of Emmy Noether. Otto Haupt held a professorship inErlangen since 1921. In his reminiscences [Hau88]5 he wrote:

Ich kam nach Erlangen als „klassisch“ gebildeter Mathematiker, nochvöllig unberührt von den damals aufkommenden, als „modern“ bezeich-neten neuen Ideen in der Mathematik. In dieser Verfassung machte ich

4Steinitz speaks of exensions of the first kind and second kind respectively. The terminology separable andinseparable appears in van der Waerden’s Moderne Algebra. It seems probable that this terminology had beencreated by Artin. For, van der Waerden reports in [vdW75] that he took lecture notes of Artin’s course on algebrain the summer of 1926. He says: “In the theory of fields Artin mainly followed Steinitz, and I just worked outmy notes … the presentation given in my book is Artin’s.”

5written when Haupt was 100 years old.


die Bekanntschaft der … eifrigen Propagandistin der modernen AlgebraEmmy Noether. Auf gemeinsamen Spaziergängen erzählte E. N. uns vonihren algebraischen Arbeiten. Ich verstand nicht viel von ihren Erzäh-lungen und fragte E. N. wie ich zu einem besseren Verständnis kommenkönne. Sie verwies mich als beste Einführung auf die 1910 erschieneneCrellearbeit von Steinitz.

I arrived in Erlangen as a “classically” educated mathematician, stilluntouched by the so-called “modern” ideas which emerged at that time.In these circumstances I got to know Emmy Noether6, … the eagerpropagandist for modern algebra. On joint walks she told us about heralgebraic work. I did not comprehend much of what she told us, and Iasked her how to get to a better understanding. She recommended theCrelle paper of Steinitz which had appeared in 1910.

We can picture the situation: Emmy Noether being an ardent walker, stridingspeedily up the Rathausberg near Erlangen and fervently persuading Haupt, whotries to keep pace with her, not only to understand modern algebra but also to write atextbook on it. Which he did.

For one year, 1929–1930, Haupt’s book was the only source for outsiders to learnabout the “modern” ideas of algebra. Even in far awayYale, a young student, SaundersMac Lane, was told by his teacher Oystein Ore to

read the monograph by Steinitz and the textbook on algebra by OttoHaupt…

As Mac Lane reports [ML81], this happened in the year 1929, one year beforevan der Waerden’s Moderne Algebra appeared. Today Haupt’s book is almost forgot-ten, although it contains some more material of Steinitz’s field theory than van derWaerden’s, in particular concerning inseparability phenomena. The fact that van derWaerden’s book was more popular than Haupt’s in terms of both its number of edi-tions and translations is probably due to the style of writing. Although sometimes it ispretended that in mathematics only the facts are important, it is a common experiencethat even in mathematics the style of writing counts.

Let us turn back to the year 1910 when Steinitz’s paper [Ste10] had appeared.Besides Emmy Noether there was another mathematician who was keenly interestedin this paper, namely Kurt Hensel in Marburg. For Steinitz had mentioned Hensel’sp-adic fields in a footnote of the introduction:

Zu diesen allgemeinen Untersuchungen wurde ich besonders durch Hen-sels Theorie der algebraischen Zahlen angeregt, in welcher der Körperderp-adischen Zahlen den Ausgangspunkt bildet, ein Körper, der weder

6At that time Emmy Noether was living in Göttingen but she used to visit her home town Erlangen from timeto time.


den Funktionenkörpern noch den Zahlkörpern im gewöhnlichen Sinnebeizurechnen ist.

I was inspired to these general investigations by Hensel’s “Theory ofAlgebraic Numbers”, in which the field of p-adic numbers is used as thestarting point. Such a field cannot be counted among the function fieldsor number fields in the ordinary sense.

Here, Steinitz refers to Hensel’s book [Hen08] which had appeared in 1908.In the year 1912 Hensel attended the 5th International Congress of Mathematicians

(ICM) in Cambridge, England. There he met the Hungarian mathematician JosefKürschak who gave a talk with the title:

Über Limesbildung und allgemeine Körpertheorie.

On the concept of limit and general field theory.

Kürschak worked in the framework of Steinitz’s abstract field theory. He definedwhat today is called a valuation of a field K, as a map a 7! jaj from K to thereal numbers, with the standard properties. He showed that Cantor’s method, whichCantor had used for the construction of the reals by means of Cauchy sequences ofrationals, works for any valuation of an abstract field in the sense of Steinitz. In thisway he constructed the completion of a valued field. As an application this gives aconstruction of Hensel’s p-adic field as the completion of the p-adic valuation of therationals. This method is standard today.

Moreover, Kürschak showed that the algebraic closure (in the sense of Steinitz)of a complete field carries a unique valuation extending the valuation of the basefield. In other words, he proved that a complete field is Henselian in contemporaryterminology. In this abstract setting Kürschak used and proved what is now calledHensel’s Lemma. It seems remarkable that his proof works at the same time for botharchimedean and nonarchimedean valuations.

Kurt Hensel was impressed by this and he took Kürschak’s paper for publicationin Crelle’s Journal of which he (Hensel) was the chief editor. The paper [Kür13]appeared in volume 142 as kind of follow-up to Steinitz’s paper [Ste10] which hadappeared in volume 137.

Kürschak did not publish any further paper on valuation theory. But there wasa young mathematician in Marburg with Hensel who was eager to take over and toextend, generalize and simplify the Steinitz–Kürschak theory. This was AlexanderOstrowski.7

Ostrowski was a brilliant young mathematician. He had started his studies in Kievbut was sent by his academic teacher Grave to Germany since, being of Jewish origin,Ostrowski seemed not to have much chance in the academic world in Russia at that

7Hensel himself, although interested in Kürschak’s work, stayed on the traditional side. He continued to usehis own construction of p-adic fields, based on p-adic power series.


time. Ostrowski arrived in Marburg in 1911 when he was 18 years old. He soon wasengrossed in valuation theory in the sense of Steinitz–Kürschak. His contributionsduring the years 1913–1918 shaped valuation theory into essentially the form we usetoday. One of the interested readers of Ostrowski’s papers was Emmy Noether, whoin the year 1916 started a correspondence with him.8

As a somewhat bizarre story let us mention that Ostrowski, being a citizen ofRussia, had been interned in Marburg during World War I when Germany was atwar with Russia. But Hensel could persuade the authorities to permit Ostrowski touse the University Library for his studies during the day. Hence for more than threeyears Ostrowski sat daily in the reading room of Marburg University Library. It wasthere where he wrote his seminal papers on valuation theory which were publishedin Crelle’s Journal, Acta Mathematica and Mathematische Annalen. He also wrotehis big monograph on valuation theory [Ost34] which was completed in 1916 butappeared only in 1934 – not as a book but in three parts in the journal MathematischeZeitschrift. Van der Waerden says in [vdW97]:

Ostrowski setzte der Bewertungstheorie mit seinen großen Abhandlun-gen in der Mathematischen Zeitschrift, Band 39, die Krone auf. DieDarstellung dieser Theorie im zweiten Band meiner Algebra beruht ganzauf dem Werk von Ostrowski.

Ostrowski crowned valuation theory with his great papers in the Mathe-matische Zeitschrift, volume 39. The presentation of this theory in thesecond volume of my Algebra is completely based on Ostrowski.9

After the war, in the year 1918, Ostrowski left Marburg for Göttingen. He did notpublish any other paper on valuation theory. (But in his paper [Ost33] on Dirichletseries he used the results of Steinitz–Kürschak and himself.)

It was some years later, in 1921, that the young student Helmut Hasse arrivedin Marburg from Göttingen in order to study p-adic fields. There Hasse learnedabout the work of Steinitz–Kürschak–Ostrowski. In the course of many years Hassebecame, in the words of van der Waerden [vdW75],

Hensel’s best and a great propagandist of p-adic methods.

In the fall of 1922 Hasse went to the University of Kiel as Privatdozent; he stayedthere until 1925. Since 1920 Steinitz had held a professorship in Kiel; thus Hasse andSteinitz were at the same university during those years. I did not find any informationabout whether Hasse and Steinitz had mathematical discussions or joint work during

8For more details, see [Roq02].9In the same paragraph in [vdW97] van der Waerden says that valuation theory was started by Rychlik. But

there seems to have been some mix-up. As said above, valuation theory had been inititiated by Kürschak in1912 on the basis of Steinitz’s paper. Rychlik has given some contributions to it, beginning in 1919, and he citesKürschak and Ostrowski.


this time. Steinitz was known as “der große Schweiger”10 (the great silent man) –which meant that it was not easy to come into close contact with him. But certainlyboth had met. Frei [Fre77] reports that there was only one office in the mathematicsdepartment, and this had to be shared by all staff : by the two professors Steinitz andToeplitz and the two Privatdozenten Hasse and Robert Schmidt.

In the academic year 1924/25 Hasse gave a course on “Höhere Algebra”. In thesecond part he presented field theory in the form of Steinitz’s paper [Ste10]. Hasse’snotes from this course became the basis for his two-volume textbook Höhere Algebra(Higher Algebra). The book appeared in the Göschen textbook series, which at thattime was well known among the German-speaking mathematicians. Its second partappeared 1927. (This was two years before the publication of Haupt’s book mentionedabove.) The referee (R. Brauer) of this part stated:

Der zweite Band behandelt die Theorie der Gleichungen höheren Grades,er schließt sich an die Arbeit von Steinitz (1910) an …

The second volume covers the theory of equations of higher degree. Itsconcept follows that of Steinitz’s paper (1910) …

In the correspondence between Hasse and Emmy Noether [LR06] one can see thatthe latter sent Hasse some advice for his algebra book. But this concerned certaindetails of proof only, e.g., the primitive element theorem. As said above, Hassedevised the concept of the book during his years in Kiel, independent of EmmyNoether.

Hasse’s estimate of Steinitz’s paper can be seen from a footnote in the secondvolume of his algebra textbook where he said:

In diese grundlegende Originalarbeit zur Körpertheorie sollte jeder Al-gebraiker einmal hineingesehen haben.

Every algebraist should have read at least once this basic original paperon field theory.

In order to facilitate this, Hasse had Steinitz’s paper reprinted in book form,together with comments and an appendix by Baer [Ste30]. This happened in theyears 1928/29 when Hasse held a professorship at the University of Halle. ReinholdBaer was Privatdozent there. In the preface of the book the editors Baer and Hassepraise the text as a

klassisch schöne, formvollendete und in allen Einzelheiten durchge-führte Darstellung…

classically beautiful, perfectly structured exposition taking care of everydetail…

10Quoted from Haupt’s reminiscences [Hau88].


And they continue:

… auch heute noch ist die Steinitzsche Arbeit eine vortreffliche, ja ger-adezu unentbehrliche Einführung für jeden, der sich auf dem Gebiet derneueren Algebra eingehenden Studien hingeben will.

… still today Steinitz’s paper is an excellent and in fact indispensableintroduction for everybody who wishes to study modern algebra moreextensively.

The appendix by Baer contains a detailed presentation of Galois theory, whichwas not explicitly covered by Steinitz.

The comments by the two editors concern those proofs in Steinitz’s paper whichuse Zermelo’s well-ordering theorem and the principle of transfinite induction. Thesewere necessary to treat infinite algebraic extensions, or fields with infinite degree oftranscendency over their prime field. Steinitz was well aware that this depends onthe axiom of choice which at that time was not generally accepted. He said:

Das Auswahlprinzip erscheint auch unvermeidlich, wenn man den Be-weis der Existenz einer algebraisch abgeschlossenen Erweiterung fürjeden beliebigen Körper führen will … Noch stehen viele Mathematikerdem Auswahlprinzip ablehnened gegenüber. Mit der zunehmenendenErkenntnis, dass es Fragen der Mathematik gibt, die ohne dieses Prinzipnicht entschieden werden können, dürfte der Widerstand gegen dasselbemehr und mehr schwinden …

Also, the axiom of choice seems to be unavoidable if one wishes toprove the existence of an algebraically closed extension of an arbitraryfield11… Many mathematicians still object to the use of the axiom ofchoice. This resistance against using the axiom of choice will dwindlewith the realization that there are mathematical problems which cannotbe decided without this axiom …

But those proofs in Steinitz’s paper which use the principle of transfinite inductionwere somewhat long-winded and therefore the comments of Baer and Hasse try tosimplify and streamline Steinitz’s arguments. Today we would prefer to use Zorn’sLemma instead; this would lead to a still greater simplification combined with aconsiderable shortening of Steinitz’s paper (which originally had 134 pages). ButZorn’s Lemma was not yet formulated in 1929.

The Hasse–Baer edition of Steinitz’s paper was reprinted 1950 by the ChelseaPublishing Company in NewYork, in its series reprinting classical treatises. (In 1997the American Mathematical Society acquired Chelsea. This title is not listed anymore

11But note that Banaschewski proved in 1992 that the existence and uniqueness of the algebraic closure canbe derived from the Boolean Ultrafilter Theorem already, which is weaker than the axiom of choice [Ban92].


as being available.) I had bought a copy myself in the 1950s and followed Hasse’sadvice to read this classic work.

Let us return to the year 1924. In that year the Hamburger Abhandlungen pub-lished the paper by Artin and Schreier: Algebraische Konstruktion reeller Körper(Algebraic construction of real fields). This paper is in some sense a companion toKürschak’s: whereas Kürschak developed the notion of valued field on the basis ofSteinitz’s abstract field theory,Artin and Schreier do the same for the notion of orderedfield. They construct the real closure of an ordered field with the help of Zermelo’swell ordering theorem – similar to what Steinitz had done for the construction of thealgebraic closure. In fact, they cite Steinitz in connection with some details of thisproof. And in the introduction the authors say:

E. Steinitz hat durch seine „ Algebraische Theorie der Körper“ weiteTeile der Algebra einer abstrakten Behandlungsweise erschlossen; seinerbahnbrechenden Untersuchung ist zum großen Teil die starke Entwick-lung zu danken, die seither die moderne Algebra genommen hat …

E. Steinitz, through his “Algebraic Theory of Fields”, has opened uplarge parts of algebra to an abstract treatment; since then, thanks to hisgroundbreaking work, modern algebra has seen a strong revival …

Nowadays we do not often find such enthusiastic references to Steinitz [Ste10] incurrent mathematical papers. The reason for this is that the main ideas and results ofSteinitz have become a matter of course, not the least through the early textbooks byHasse, Haupt and van der Waerden mentioned above.

At the end of his introduction Steinitz says in his paper of 1910:

Der vorliegende Aufsatz behandelt nur die Grundzüge einer allgemeinenKörpertheorie. Weitergehende Untersuchungen sowie Anwendungen aufGeometrie, Zahlen- und Funktionentheorie beabsichtige ich, in einigenweiteren Anhandlungen folgen zu lassen.

The present article covers the foundations of a general field theory only.I am planning to follow-up with more advanced investigations, and withapplications to Geometry, Number Theory and the Theory of Functions.

But Steinitz did not publish anything in this direction, and also in his literary estatenothing of this kind was found. We do not know why he did not later write what he hadannounced. In any case, we observe that his 1910 paper has exerted a great influenceupon the delopment of modern algebra during the next decades. As evidence for thiswe may cite the following papers which are to be considered as immediate follow-upsto [Ste10], and have each opened up a long line of development:

• Analysis, including p-adic analysis: Kürschak–Ostrowski on valuation the-ory [Kür13], [Ost34];


• Number Theory: Emmy Noether on Dedekind domains in abstract fields[Noe26];

• Algebraic Geometry: Emmy Noether on primary decomposition of ideals in aNoetherian ring [Noe21], and van der Waerden on the foundations of algebraicgeometry [vdW75];

• Real Algebra: Artin and Schreier on real fields [AS27a], [AS27b].

This, of course, is not a complete list. 12 For a biography of Steinitz we refer tothe Dictionary of Scientific Biography.

Ackowledgment. I am indebted to Keith Conrad for helpful critical comments, andfor streamlining my English.

12Compare with [Pur73] and [Kle99].

Chapter 10

Heinrich-Wolfgang Leopoldt

Obituary.

Journal of Number Theory 132 (2012), 1641–1644.

H. W. Leopoldt

Heinrich-Wolfgang Leopoldt, member of the editorial board of Journal of NumberTheory from 1969 to 1987, has passed away on July 28, 2011 at the age of almost84 years. His name will be remembered as the discoverer of the p-adic L-functions( jointly with T. Kubota) and as the author of the Leopoldt conjecture.

Leopoldt was born and raised in the small German town of Schwerin at the Balticsea. Still a school boy he was drafted in war time to various military services. Afterthe war he started an apprenticeship in view of the uncertain prospects of the future.However there was a former mathematics teacher with whom Leopoldt played music,and who introduced him also to astronomy and its mathematical fundamentals. Hesuggested to Leopoldt to finish his Gymnasium (high school) in order to be able toattend university. Leopoldt did so, and in the academic year 1947/48 he started hismathematical studies at the Humboldt University in Berlin.

240 10 Heinrich-Wolfgang Leopoldt

One of the first lecture courses he attended was an introduction to number theoryby Helmut Hasse. In his Erinnerungen (memoirs) Leopoldt reports that this lecturehad left a deep impression upon him, in particular Hasse’s remarks on the relationshipbetween beauty and truth in mathematics, combined with various instances of parallelsbetween number theory and music. This lecture, Leopoldt recalls, led him to studynumber theory with highest priority. We can observe this dedication to number theorythroughout his mathematical work.

In the year 1950 Hasse left Berlin for Hamburg and his student Leopoldt followedhim there. 1954 he obtained his Ph.D. at Hamburg University. Thereafter he got aposition as assistant professor in Erlangen until 1962, interrupted for two years atthe Institute for Advanced Study in Princeton. 1959 he made his Habilitation, i.e.,his qualifying examination as a university teacher. 1962–1964 he taught at TübingenUniversity, again interrupted by a visiting professorship at Johns Hopkins Universityin Baltimore. In the year 1964 he obtained an offer for a permanent position at John’sHopkins and another one at Karlsruhe University. He accepted the latter.

Leopoldt’s mathematical work is governed by the attempt “to systematically openup the structure of abelian fields such that one can work in them freely in the sameway as one can work in quadratic fields.” This challenge had been formulated byHasse in a monograph 1952, and his student Leopoldt set about to perform this taskstep by step. Here I cannot give an assessment of Leopoldt’s complete work. Butwhile I am writing this obituary I remember particularly three of his most brilliantachievements which had impressed me at the time – and I would like to share myrecollections here.

First I remember his paper on the structure of the ring of integers in an abelian fieldK which was his Habilitationsschrift at Erlangen [Leo58]. If K is tamely ramifiedthen Emmy Noether had shown that there exists a integral normal base, i.e., the ringof integers I ofK isG-module isomorphic to the group ring QŒG�, whereG denotesthe Galois group of K. However, if the ramification is not tame then there does notexist a normal integral base and it was not known how Noether’s theorem shouldbe be modified in order to describe the G-structure of the integers. But Leopoldt’spaper solves this problem, first in an abstract setting showing that I is G-module-isomorphic to a certain order of QŒG�, and then interpreting this in a very explicit andconcrete manner by constructing explitly a normal basis with the help of the Gaussiansums �./ belonging the characters of G.

Secondly I recall his brilliant paper, composed in Princeton, on the Spiegelungs-satz in an absolute Galois number fieldK [Leo58]. This concerns the structure of theclass group C ofK, more precisely: the structure of its p-part Cp where p is a fixedprime. Assume that the degree ofK is prime to p and that the p-th roots of unity arein K. The action of the Galois group G of K induces a direct decomposition of Cp

into subgroups C' where ' ranges over the p-adically irreducible characters of G.Let 0 denote the character belonging to the representation ofG on the p-th roots of


unity. Then there is what Leopoldt calls the reflection map ' ! x' with

x'.�/ D 0.�/'.��1/ .for � 2 G/:

This construction of x' had arisen in various situations before but Leopoldt had dis-covered a relationship between the '-part C' and the x'-part Cx' . In some sense bothare of about the same size. More precisely, if e' , ex' denote the G-ranks of C' , Cx'respectively then

�ıx' � ex' � e' � ı' :

where the bounds ı' , ıx' depend in a well defined manner on the p-adic Galoisstructure of the group of units E ofK. Hence the study of the Galois structure of theunits of K leads to information about the deviation of the rank of C' from that of itsmirror image Cx' .

Leopoldt’s proof of this theorem uses class field theory; it is quite lucid andstraightforward. It did not use any newly developed method and could have beenproved much earlier. The essential new ingredient is the way of looking at the variousobjects from the structural point of view. Instead of dealing with class numbersLeopoldt deals with the class groups and their Galois structure. This point of viewhad been advocated by Hasse in his monograph cited above, and Leopoldt now usesit competently to advance a big step towards the understanding of the class groups ofnumber fields.

Nowadays this point of view, i.e., the investigation of class groups by means of Ga-lois action, has become standard in algebraic number theory, as it is manifested, e.g.,in what is called the Iwasawa theory. At the time of publication the Spiegelungssatzwon widespread interest among number theorists – not only because it offered a com-mon background for classically known results about divisibility properties of classnumbers, but also since it admitted to obtain much more information of this kind.Whereas classically (since Kummer) the properties of class numbers are connectedwith the Bernoulli numbers Bn, Leopoldt’s new results referred to his generalizedBernoulli numbers Bn

belonging to Dirichlet characters . They are defined recur-sively in a similar way as are the ordinary Bernoulli numbers. Leopoldt showed thathis Bm

, which are contained in the cyclotomic field Q./, occur as the values of theDirichlet L-functions L.s; / at the odd negative integers:

L.1 �m;/ D �Bm

m:

These relations lead to the third oustanding result of Leopoldt which I wouldlike to recall, namely the discovery (jointly with Kubota) of the p-adic L-functions[KL64]. They belong to the standard tools of today’s algebraic number theory butperhaps it is good to recall that they are due to Leopoldt. The story is as follows.

For any prime p his generalized Bernoulli numbers Bm satisfy certain p-adic

congruences, the so-called Kummer congruences. Leopoldt observed that those con-

gruences can be interpreted such that theBm

�

mare in a sense p-adically continuous

242 10 Heinrich-Wolfgang Leopoldt

functions of m. More precisely: There is one and only one p-adically continuousfunction Lp.s; / defined on Zp such that

Lp.1 �m;/ D �Bm

m.1 � .p/pm�1/

for the negative integers 1 �m with m 0 mod p � 1. 1 These numbers 1 �m aredense in Zp . But it turns out that Lp.s; / is holomorphic in a region which is largerthan Zp , at least if ¤ 1 whereas for D 1 there is one pole for s D 1.

Based on these p-adic L-functions Leopoldt considered for any abelian numberfieldK the corresponding p-adic zeta function �K; p .s/ as the product of theLp.s; /

for the characters of K. He arrives at p-adic class number formulas which are theanalogue of the ordinary class number formulas for abelian fields K, and they lookquite similar but the terms have to be interpreted in the p-adic sense.

However there was one obstacle to overcome, namely the proof of non-vanishingof the so-called p-adic regulator of an abelian field which appears in those formulas.This regulatorRK; p is obtained if one replaces the ordinary logarithms in the classicalregulator by thep-adic logarithms. The non-vanishing ofRK; p means that thep-adicrank of the group of units ofK equals its ordinary rank, i.e., r1 C r2 � 1 where r1, r2are the numbers of real or complex infnite primes respectively.

Leopoldt had tried hard to prove this conjecture. He delayed the publication of thesecond part of his L-series paper [Leo75] since he still hoped to be able to include aproof of it. Nevertheless he presented an exposition of his theory when he lectured atJohns Hopkins in the year 1964. In this form the theory somehow made it to Prince-ton where Iwasawa included it in his Lectures on p-adic L-functions. This madeLeopoldt’s theory widely known and started an extended research in consequenceof which p-adic L-series became indispensable tools of number theorists. Brumersucceeded 1967 to prove Leopoldt’s conjecture for arbitrary abelian fields.2

A characteristic feature of Leopoldt’s work is that he aims at concrete results givenby explicit and effective formulas. He competently uses and investigates abstractstructures, but such considerations serve him as motivation only, as guideline towardsthe goal of explicit algorithms. To a large degree his results were obtained by extendednumerical computations.

This attitude led him quite early to develop computer programs for the use inalgebraic number theory. His team in Karlsruhe was one of the first in Germany whichsystematically developed the necessary algorithms for this project, also in cooperationwith Hans Zassenhaus. A number of Leopoldt’s students are now working in scientificcomputing and so follow up his ideas.

Leopoldt’s personality can be characterized as quiet, unassuming, always willingto hold back his own in favor of supporting the cause. His firm and objective counsel

1For simplicity we assume here that is an even character and that p > 2.2Recently Mihailescu has announced a proof of Leopoldt’s conjecture for all number fields.


in scientific matters, always to the point, was valued by all who had to deal with him.His lectures stood out by their clarity and intensity. He was known as a master inexposition. The list of his publications is not large but his work belongs to the pearlsof mathematical research in the last century.

Leopoldt was married and has five children. After his retirement from the Uni-versity of Karlsruhe he had moved to a small village in Northern Germany where hededicated himself to his beloved piano music.

Chapter 11

On Hoechsmann’s Theorem

On the local-global principle for embedding problems over global fields.Comments to an old paper of Klaus Hoechsmann.

Israel Journal of Mathematics 141 (2004), 369–379.

11.1 Introduction 24511.2 Statement of the result 24511.3 The setting 24711.4 Hoechsmann’s theorem 24911.5 Construction of counter examples 252

11.1 Introduction

The origin of this note was the attempt to answer a question of Moshe Jarden whohad asked me:

Hat jeder Zahlkörper ein endliches Einbettungsproblem, das lokal über-all lösbar aber global nicht lösbar ist?

Does every number field admit a finite embedding problem which islocally solvable everywhere but not globally solvable?

As a first reaction I referred him to an old paper of Hoechsmann [Hoe67] on theembedding problem. But after a second reading of Hoechsmann’s paper I found thatthe answer to Moshe’s question – which is affirmative – is not explicitly stated there.Although the answer can be readily derived using Hoechsmann’s ideas, it is perhapsnot without interest to do this explicitly. This is what I propose in this note.

11.2 Statement of the result

Let K be a global field and GK its absolute Galois group. Let A be a finite GK-module. The action of GK on A factors through a finite factor group. Let G be such

246 11 On Hoechsmann’s Theorem

a factor group, i.e. G D GK=U where U is an open normal subgroup of GK whichacts trivially on A. We consider embedding problems of the form

GK

'

��

ˆ

��

��

1 �� A �� E �� G ��

��

1

1

(11.1)

where ' is the natural projection, and where E is a group extension of A with G.Such group extensions correspond to the cohomology classes " 2 H 2.G;A/. Weare looking for solutions ˆ of the embedding problem. We do not require that ˆ besurjective. But note that for a global field, it is well known that the existence of anysolution, surjective or not, implies the existence of a surjective solution. (A proof canbe found in Hoechsmann’s paper.)

If p is a prime of K then Kp denotes its completion. The absolute Galois groupGKp is considered as a subgroup ofGK , viz. the decomposition group of an extensionof p to the algebraic closure (unique up to conjugation). Let Gp D '.GKp/ denotethe decomposition group of p in G.

Given an embedding problem (11.1) its localization at p is

GKp

��

��

1 �� A �� Ep �� Gp ��

��

1

1

(11.2)

where Ep is the inverse image of Gp under the map E ! G. The factor system ofthis localization is the restriction ResGp."/ of the factor system " of (11.1).

The solvability of (11.1) implies the solvability of (11.2) for each p. The “Local-Global Principle” LGP.A;K/ asserts that conversely, if an embedding problem (11.1)is locally solvable for each p then it is globally solvable. For a given GK-module Athe Local-Global Principle may hold or may not hold. Our result in this note is

Theorem 6. For every global fieldK of characteristic ¤ 2 and any givenm � 3 thereexists a GK-module A of order 2m such that the Local-Global Principle LGP.A;K/does not hold.

Remark. The modules A to be constructed will be cyclic groups of order 2m withthe action of GK defined suitably. If 2 is replaced by a prime number p > 2 then


the situation is completely different. For, ifGK acts on a cyclic group A of order pm

with p > 2 then the Local-Global Principle LGP.A;K/ does hold (irrespective ofthe characteristic of the field K). This is a consequence of Gudrun Beyer’s theorem.(See Corollary 11 below.) The exceptional role of the prime 2 in this context is aconsequence of the difference in the structure of the automorphism group of cyclicgroups ofp-power orderpm. Ifp > 2 then the automorphism group is cyclic whereasif p D 2 this is not the case for m � 3. In this respect the situation here is similar tothe situation of the Grunwald–Wang theorem. (See [LR03].)

Concerning the characteristic hypothesis in Theorem 6, this is necessary if onewishes to construct counter examples to the Local-Global Principle by means ofcyclic groups A, as we do in this paper. If K is of characteristic 2 and A is a cyclicgroup of 2-power order with any action of GK then the LGP.A;K/ holds. Thisis a consequence of Witt’s theorem that for a global field K of characteristic 2 themaximal pro-2-factor group of GK is free in characteristic 2 (and similarly for anynon-zero characteristic). I do not know whether non-cyclic groups A can serve ascounter examples to the Local-Global Principle.

11.3 The setting

Let me first recall some of the results in Hoechsmann’s paper.The solvable embedding problems (11.1) form a subgroup ofH 2.G;A/, and this

is precisely the kernel of the inflation map

inf W H 2.G;A/ ! H 2.GK ; A/: (11.3)

(Note that the inflation map is well defined since the kernel ofGK ! G acts triviallyon A.) This holds for any base field K, hence also for the localizations. Now, everyelement in H 2.GK ; A/ is the inflation of some element in H 2.G;A/ for a suitablefinite factor group G. We conclude:

Proposition 7. The Local-Global Principle LGP.A;K/ holds if and only if the map

H 2.GK ; A/h ��

QpH

2.GKp ; A/ (11.4)

is injective.

At this point Hoechsmann cites the duality theorem of Tate–Poitou for globalfields. That duality theorem holds if the order of A is relatively prime to the char-acteristic of K (including the case of characteristic 0) which we assume henceforth.Let yA denote the dual GK-module of A. It consists of the characters of A, i.e.,the homomorphisms of A into the multiplicative group of the algebraic closure ofK.


The action of GK on yA is given by

� .a/ D �.a��1

/��

.a 2 A; � 2 GK/: (11.5)

Note that in this formula � acts twofold: First ��1 acts onA sinceA is aGK-module.Secondly, � acts on the character values since � is an automorphism of the algebraicclosure of K. In Hasse’s terminology, this is a “crossed action” of GK on yA.

Now, the Tate–Poitou duality theorem asserts that for a global fieldK, the map hin (11.4) is dual to the following map:

H 1.GK ; yA/ j �� QpH

1.GKp ;yA/: (11.6)

In particular, h is injective if and only if j is injective. We obtain:

Corollary 8. The Local-Global Principle LGP.A;K/ holds if and only if the map jin (11.6) is injective.

By this result, the problem is transferred from cohomological dimension 2 todimension 1. This is the starting point of Hoechsmann. First he reduces the problemto a finite factor group of GK .

Proposition 9. LetG be the action group of theGK-module yA, i.e., the factor groupof GK modulo the normal subgroup which fixes yA elementwise. Then LGP.A;K/holds if and only if the map

H 1.G; yA/ jG �� QpH

1.Gp; yA/ (11.7)

is injective.1

Here,Gp denotes the decomposition group of p inG, i.e., the image ofGKp inG.

Proof. (i) First we consider the case when G D 1, i.e. GK acts trivially on yA. Inthis case it is asserted that the LGP.A;K/ holds, i.e., that the map j in (11.6) isinjective. Now, in case of trivial action we haveH 1.GK ; yA/ D Hom.GK ; yA/. Everyhomomorphism f W GK ! yA factors through a finite, abelian factor group xG ofGK .Let N� 2 xG. Using Chebotarev’s density theorem we conclude that there exists aprime p of K whose decomposition group contains N� . Hence, if f vanishes on alldecomposition groups then f . N�/ D 0. Since this holds for all N� we conclude f D 0.

(ii) Now consider the general case. Let L be the finite Galois extension of Kcorresponding toG, so thatG is the Galois group ofLjK. Consider the commutative

1This proposition and the following corollaries remain valid for any finite factor group G of GK modulo anormal subgroup which acts trivially on yA.


diagram

0 �� H 1.G; yA/ inf ��

jG

��

H 1.GK ; yA/ Res ��

j

��

H 1.GL; yA/jL

��

0 �� QpH

1.Gp; yA/ �� QpH

1.Gp; yA/ Res �� QpH

1.GL;p; yA/(11.8)

with self-explaining notations. The rows are exact. The vertical arrow jL on the righthand side is injective by (i), forGL acts trivially on yA. Consequently, if the arrow jG

on the left hand side is injective then j in the middle is injective too, and conversely.

Corollary 10. As in Proposition 9 let G denote the action group of GK on yA. If thegroup indices ŒG W Gp� of the decomposition groups have greatest common divisor 1then LG.A;K/ holds.

For, let c 2 H 1.G; yA/. If c vanishes at p, i.e., if ResGp.c/ D 0 then it followsŒG W Gp� � c D 0. If this holds for all p then c D 0, provided the indices ŒG W Gp�

have greatest common divisor 1.

Corollary 11. If the action group G of GK on yA is cyclic then LGP.A;K/ holds.

For, if G is cyclic then by Chebotarev’s density theorem there exists p withGp D G.

Corollary 11 is the theorem of Gudrun Beyer. It is remarkable that the validityof LGP.A;K/ depends on the action of GK on the dual yA, not on A itself. This hasbeen discovered by Gudrun Beyer. For Corollary 10 Hoechsmann cites Demuškinand Šafarevic.

11.4 Hoechsmann’s theorem

From now on we assume thatA is a cyclic group. After decomposingA into its Sylowcomponents we may assume that the order ofA is a prime power, jAj D pm. Its dualyA is also a cyclic group and j yAj D pm too. If p > 2 then the automorphism group ofyA is cyclic and it follows thatG is cyclic, hence LGP.A;K/ holds by Gudrun Beyer’s

theorem (Corollary 11).Consequently, in looking for a counter example to LGP.A;K/ we have to take

p D 2. (This implies thatK is of characteristic ¤ 2 since the order of A is supposedto be relatively prime to the characteristic ofK.) TheGK-moduleA should be a cyclicgroup such that the action group G on yA is non-cyclic. In particular m � 3. If thereexists a prime p of K with Gp D G then by corollary 10 we have that LGP.A;K/holds. We conclude:


Let A be a GK-module which is a cyclic group of prime power order pm. If theLocal-Global Principle LG.A;K/ does not hold then the following conditions aresatisfied:

1. p D 2.2. The action group G of GK on yA is non-cyclic, hence m � 3.3. For every prime p of K, the decomposition group Gp is a proper subgroup

of G.

Now we can formulate Hoechsmann’s theorem:

Theorem 12. The conditions 1–3 above are not only necessary but also sufficient forA to be a counter example to LGP.A;K/.

In view of Proposition 9 this is an immediate consequence of the following grouptheoretical observation. For simplicity we write X instead of yA.

Lemma 13. LetX be a cyclic group of order 2m .m � 3/ andG a non-cyclic groupof automorphisms ofX . Then there exists 0 ¤ c 2 H 1.G;X/ such that its restrictionResH .c/ vanishes for every maximal subgroupH ¤ G.

Proof. We identifyX D Z=2m (additively) andG with a group of units in .Z=2m/�.The action of G on X is given by multiplication. Any element in H 1.G;X/ can berepresented by a crossed homomorphism f W G ! X . The functional equation of acrossed homomorphism is

f .��/ D �f .�/C f .�/ for �; � 2 G: (11.9)

In particular, for � D � we note that

f .�2/ D .� C 1/f .�/: (11.10)

We shall prove the lemma by explicitly exhibiting a crossed homomorphism f rep-resenting c.

The non-cyclic group G is a direct product

G D h�1i huiwhere u ¤ 1 is a certain unit of X which can be assumed to be u 1 mod 4. (Ifthis should not be the case then we replace u by �u.) Let k be the exact exponent bywhich 2 appears in u � 1, so that

u � 1 D 2k

where is not divisible by 2, hence a unit in X . We have

2 � k � m � 1:


(If k would be � m then u 1 mod 2m, contradicting the fact that u ¤ 1 as operatoron X .) The group theoretical meaning of k is the following:

The group 2m�kX consists precisely of those elements ofX which are fixed by u.For, the relation ux x mod 2m is equivalent to .u � 1/x 0 mod 2m which,

by definition of k, means x 0 mod 2m�k .Every crossed homomorphism f W G ! X is already determined by its values

on the generators �1 and u ofG. We claim that there is a crossed homomorphism f

with the valuesf .�1/ D 2m�k; f .u/ D 0 (11.11)

and that its class c 2 H 2.G;X/ satisfies the requirements of the lemma.First we consider the subgroup h�1i of G of order two. Consider the function

f0 W h�1i ! X given by the values f0.�1/ D 2m�k; f0.1/ D 0. This is a crossedhomomorphism. To verify this one has to check the validity of (11.10) for � D �1only. Indeed, we have

f0..�1/2/ D .�1C 1/2m�k D 0 D f0.1/:

We have the exact sequence

1 ! hui �! G �! h�1i ! 1:

As observed above, the value f0.�1/ D 2m�k is fixed by u. Hence we may extendf0 W h�1i ! X by inflation to a crossed homomorphism f W G ! X such thatits values f .�/ depend on the residue class of � modulo hui only. This crossedhomomorphism satisfies (11.11).

Let c 2 H 1.G;X/ denote the class of f . We claim that the restriction of c toevery maximal subgroup of G vanishes. There are three maximal subgroups of G,namely the two cyclic groups hui and h�ui, and the group h�1; u2i which in generalis not cyclic except if u2 D 1 (which means that k D m � 1).

The restriction of c to hui vanishes since f .u/ D 0 by (11.11).As to the restriction of c to h�ui we first note that f .�u/ D f .�1/ D 2m�k

does not vanish. But consider a crossed homomorphism g W G ! X belonging to thesame class c as f , which means that

g.�/ D f .�/C .� � 1/x .� 2 G/ (11.12)

for some x 2 X . Can we choose x 2 X such that g.�u/ D 0 ? This means

f .�u/ D 2m�k D �.�u � 1/x D .uC 1/x:

Since u 1 mod 4 we have uC 1 2 mod 4 hence uC 1 D 2� with � a unit inX . Hence by choosing x D ��12m�k�1 we indeed have g.�u/ D 0.


Can we choose x such that g vanishes on the third maximal group h�1; u2i ? Thismeans, firstly, g.�1/ D 0 and thus

f .�1/ D 2m�k D �.�1 � 1/x D 2x (11.13)

and so we take x D 2m�k�1. Secondly, the condition g.u2/ D 0 requires that

f .u2/ D 0 D �.u2 � 1/x D �.u � 1/.uC 1/x D �� 2kC1 � x:The same x D 2m�k�1 as above satisfies this condition since 2mx D 0.

We have now shown that c vanishes if restricted to any of the three maximalsubgroups of G. It remains to verify that c ¤ 0 in H 1.G;X/. In other words: Itis not possible to choose x 2 X such that g.�1/ D g.u/ D 0. Now the conditiong.�1/ D 0 implies by (11.13) that x is precisely divisible by 2m�k�1 (and not by ahigher power of 2). On the other hand, the condition g.u/ D 0 requires that

f .u/ D 0 D �.u � 1/x D � � 2k � xand hence x should be divisible by 2m�k . Both these conditions are not compatible,and so c ¤ 0.

11.5 Construction of counter examples

In the following we let A be a cyclic group of order 2m withm � 3. We try to definea non-cyclic action ofGK onA such that condition 3 of Theorem 12 is satisfied. Thiswill give a counter example to LGP.A;K/. The main tool for this is the following

Lemma 14. For any global field K there exists an abelian extension LjK of pre-scribed 2-power degree 2rC1 whose Galois group G D Gal.LjK/ has the structure

G � Z=2 Z=2r ;

and such that for every prime p ofK its decomposition groupGp is a proper subgroupof G.

There are many possibilities to construct such a field extension. First assume thatK is a number field. Consider the field K.2/ of 2-power roots of unity over K. ItsGalois group is either a free cyclic pro-2-group (for instance if

p�1 2 K) or else itis the direct product of such a group with a group of order 2. In any case the Galoisgroup ofK.2/jK contains finite cyclic factor groups of arbitrary large 2-power order.Accordingly let L0jK be a cyclic extension of degree 2r which is contained in K.2/.We observe that the only primes p of K which are ramified in L0 (if there are any)are divisors of 2. This follows from the fact that 2 is the only prime number in Qwhich is ramified in Q.2/.


Now we take a rational prime number p > 2 such that

p 1 mod 2N (11.14)

for sufficiently large N and put

L D L0.pp/:

If N and hence p is sufficiently large then p is unramified in K, i.e., every primedivisor pjp appears in p with the exponent 1. We conclude that

pp … K, and that

p is ramified in the quadratic extension K.pp/. Therefore K.

pp/ is not contained

in L0, and K.pp/ is linearly disjoint to L0 over K. The Galois group G of LjK is

the direct product of Gal.L0jK/ (which is cyclic of order 2r ), with Gal.K.pp/jK/

(which is of order 2).Let p be a prime of K and Gp its decomposition group in G. If p is unramified

in L (including the case when p is an infinite prime) then its decomposition groupis cyclic and hence Gp is a proper subgroup of G. If p is ramified in L then eitherpj2 or pjp. In the first case, pj2, if N � 3 then (11.14) implies

pp 2 Q

2, hencep

p 2 L0;p, thus Lp D L0;p is of degree � ŒL0 W K� D 2r over Kp. Hence itsGalois group Gp is of order � 2r and thus a proper subgroup of G. In the secondcase, pjp, let N be large enough such that L0 is contained in the field of 2N -th rootsof unity over K. The condition (11.14) implies that Qp contains the 2N -th roots ofunity, thus L0 � Qp � Kp and consequently Lp D Kp.

pp/ is of degree � 2.

Now assume that K is a function field of characteristic ¤ 0. Let k be its fieldof constants, and consider the unique extension k0 of degree 2r over k. We putL0 D Kk0 ; this is the constant field extension of K of degree 2r . It is cyclic andunramified over K. Now let t 2 K be a separating variable. Consider a primepolynomial p.t/ 2 kŒt � with the condition that its residue field contains k0. Thiscondition is the analogue to condition (11.14) in the number field case. Since thereare infinitely such polynomials we may assume that p.t/ is not ramified in K.

If the characteristic of K is ¤ 2 then we put again

L D L0.pp.t/ /:

Quite analogous to the number field case it is seen that L satisfies the requirementsof the lemma. The situation here is even easier since L0jK is unramified, hence it isnot necessary here to discuss the prime divisors which are ramified in L0, as we hadto do in the number field case. The only primes p of K which are ramified in L arethe prime divisors of p.t/. For any such p its residue field contains k0 and hence itscompletionKp too contains k0. It follows thatKp containsKk0 D L0 and thereforeLp D Kp.

pp.t// is of degree � 2.

If the characteristic of K is 2 then K.pp.t// is inseparabel and useless for our

construction. Instead of a square root we have to use a root of the appropriate Artin–


Schreier equation:

L D L0.˛/; ˛2 � ˛ D 1

p.t/:

Again, the only primes of K which are ramified in L are the prime divisors of p.t/and the discussion now proceeds as in the case of characteristic ¤ 2.

Lemma 14 is proved. In that lemma we have not excluded the case of characteristic2 because it is not necessary. However, in the following proof we have to assume thatchar.K/ ¤ 2 in order to be able to apply Hoechsmann’s theorem which is based onthe Tate–Poitou duality theorem.

Proof of Theorem 6. Let us put X D Z=2m. The automorphism group Aut.X/consists of the units in Z=2m which act by multiplication. Aut.X/ is non-cyclic andhas the structure

Aut.X/ � Z=2m�2 Z=2:

We see that Aut.X/ is isomorphic to the Galois group G D Gal.LjK/ of the fieldextension of Lemma 14 if in that Lemma we take r D m � 2.

Let is fix an isomorphism G � Aut.X/. In this way X becomes a G-module. Xappears as a GK-module via the projection GK ! G. The action group of GK on Xis G.

Now we take A D yX . Then A is a GK-module of the same order 2m as X . We

have yA D yyX D X . Thus the action group of GK on yA is G. The conditions 1–3of Theorem 12 are satisfied in view of Lemma 14. We conclude that A is a counterexample to LGP.A;K/.

Problem. Prove Hoechsmann’s theorem directly, without reference to the Tate–Poitou duality theorem. It seems that the reciprocity law for global fields will besufficient.

Acknowledgements

The author and the publisher wish thank Elsevier, Springer Science + BusinessMedia,Verlag Vieweg + Teubner, Walter de Gruyter, the DMV, the LMS, and the ManagingEditor of the Israel Journal of Mathematics for granting permission to republish thearticles in this collection in their original or a revised and extended form. Thanksalso go to the Mathematisches Forschungsinstitut Oberwolfach for providing a highresolution copy of the photo showing Otto Grün on page 79.

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Name Index

Abel, 226Albert, 3, 29, 30, 44, 51–53, 56–73, 75,

192, 193, 210, 212Albert, Nancy, 3, 71Alexander, 145Alexandrov, 54, 119, 135, 144, 145,

148, 228Amitsur, 50Aravire, 209Archibald, 54, 55Archimedes, 178, 183Arf, 189–226Artin, 3, 5, 20, 25, 27, 31, 39, 41,

43–45, 54, 61, 74, 75, 85–87,91, 94, 112, 134, 143, 145,148, 152, 186, 190, 215,224–226, 228, 230

Bach, 126Baer, 103, 106, 112, 113, 234, 235Baeza, 209, 210, 216, 221Banaschewski, 235Bannow, 171Bartels, 116Beaumont, 123, 159Becher, 189Bernoulli, 177Bertini, 186Beyer, 247, 249Bieberbach, 116, 153Bilharz, 112Birkhoff, 144, 158, 172, 173Blumenthal, 34Bohr, 151, 164–166Bourbaki, 47, 142, 227Brahana, 89Brandt, 41, 54, 55, 145, 152, 171Brauer, 1–76, 79, 120, 122, 127, 155,

190, 192, 211, 234

Brouwer, 135, 136Brüning, 156Brussel, 50Burckhardt, 41Burnside, 91, 92, 95, 106

Cantor, 232Carleman, 107Cartan, 103Cauchy, 177Chevalley, 15, 27, 35, 44, 72, 76, 86,

87, 187, 224Chow, 171Conrad, v, 3, 189, 237Courant, 152, 155Curtis, 46

Dauben, 183, 184Davenport, 61, 171Dechamps, 171Dedekind, 99, 124, 141, 153, 228Dehn, 89Demuškin, 249Derry, 171Deuring, 17, 24, 54, 55, 69, 75, 110,

118Dick, 122, 130, 133Dickson, 4, 5, 17, 40, 41, 51, 57–59,

71, 72, 212, 215Dieudonné, 112, 114Dirichlet, 99Draxl, 210, 216, 221Duggan, 174Dukas, 157, 158

Eichelbrenner, 40Eilenberg, 103Einstein, 136, 137, 156–160, 162Eng Tjioe Tan, 26

274 Name Index

Engström, 24Euler, 177

Faltings, 188Feit, 53Fermat, 82Fischer, 133, 228Fitting, 54, 55Flanders, 161, 162Flexner, 123, 147, 150, 153, 155, 159Franz, 73, 186Frei, 164, 195, 234Frobenius, 47, 142Furtwängler, 12, 55, 87, 94, 151, 165,

166

Garibaldi, 189Gaschütz, 78Geyer, 26Gilmore, 186Göbel, 152Gordan, 132–134Grave, 232Grün, 77–116Grunwald, 23, 25

Hahn, O., 152Hall, Ph., 105, 107, 113Hardy, 151, 164, 165Hasse, 1–226, 233–236, 240, 241, 248Haupt, 176, 230, 231, 234, 236Hausdorff, 135Hecke, 74, 86, 226Heegner, 110Heisenberg, 120, 121Hensel, 3, 6, 9, 10, 29, 33, 40, 42, 44,

53, 79, 119, 139, 140, 187,231–233

Herbrand, 15, 54, 55, 87Herglotz, 195Hering, vHermann, 117, 121, 122, 125, 126Hey, 73–76, 170

Higman, 107Hilbert, 12, 69, 87, 99, 122, 132–134,

136–140, 148, 157, 158, 186,226, 228

Hoechsmann, 245–254Hoffmann, 189Holzer, 88Hopf, H., 144, 145Huppert, 78, 97, 103

Ikeda, 216Iyanaga, 27, 94, 224

Jacob, 209Jacobson, 143Jarden, 245Jehne, 109Jensen, 26

Kani, 188Kaplansky, 68Kersten, 192, 196, 204Kiepert, 181, 182Kimberling, 130, 144, 157, 161, 172,

174Klein, 133, 134, 137, 157Klingenberg, 204Knauf, 171Knus, 193, 204Köthe, 21, 22, 54, 55Koreuber, 46Kostrikin, 107Krickeberg, 109Kronecker, 81, 177Krull, 54, 178, 186, 187Kummer, 29, 31, 42, 80–83, 85, 87, 99,

139Kurosch, 114Kürschak, 232, 233

Lagrange, 177Lamprecht, 109Langlands, 215, 223–226

Name Index 275

Lasker, 228Lefschetz, 144, 145, 150, 154, 155,

158, 162, 172, 174Lehr, 118, 122, 123, 127, 172Leibniz, 175, 177, 178, 183, 185Lemmermeyer, 81, 88Leopoldt, 112, 239–243Levi, 113Levitzky, 20Lie, 114Litvinov, 159Lorenz, 3, 4, 13, 31, 75, 170, 189, 216Luxemburg, 175

Mac Lane, 139, 147, 148, 231Macaulay, 228Magnus, 89, 94, 104–108, 111, 114Mahler, 187Meitner, Lise, 152Mertens, 145Mills, 26Ming-chang Kang, 26Minkowski, 18, 139Moore, 122Mordell, 188Morton, 26Mozart, 126Müller, G., 184

Nelson, 126Neugebauer, 69, 168Neumann, B. H., 114Noether

Emmy, 1–76, 79, 80, 95, 117–174,190, 192, 195, 210, 211,224–226, 228, 230, 231, 233,234

Fritz, 123, 134, 136, 159Gottfried, 159Hermann, 159Max, 124, 130, 134Otto, 118, 122, 126, 156

Önder, 214Ore, 231Ostrowski, 232, 233

Park, 118, 122, 123, 126, 127, 147,156, 158, 173

Perron, 152, 165, 166Purkert, 229

Rella, 152, 165Ribenboim, 81, 83Robinson, 175–188Rohrbach, 84, 211Roth, 188Rychlik, 233

Saltman, 31Schappacher, 164Schilling, 210Schmeidler, 138Schmid, H. L., 101, 109–112, 196Schmidt, Erhard, 109, 133Schmidt, F. K., 54, 143Schmidt, Robert, 234Scholz, A., 88, 102Schouten, 152, 166, 167Schreier, 94Schur, 13, 17, 28, 44, 54, 94, 97, 111Schwarz, 171Scorza, 114Segre, 152, 166Sertöz, 194, 202Šafarevic, 249Shoda, 152, 166Siegel, 18, 29, 152, 166–168, 187Siegmund-Schultze, 115, 156, 159Snail, 215Speiser, 41, 42, 44, 54, 102, 106, 152,

166Spengler, 153Stauffer, 156–158, 161Steinitz, 134, 187, 227–237

276 Name Index

Süss, 107Suzuki, 108Sylow, 77

Takagi, 43, 80, 87, 152, 166, 167, 226Tate, 25, 31, 45, 143Taussky, 51, 118, 122, 126, 134, 147Teichmüller, 101Teichmüller, 201, 202Thompson, 97Tinsdale, 144Tobies, 54, 171Toeplitz, 21, 234Tollmien, 130, 132–134, 137, 164,

168–171Tornier, 116, 169, 170Tsen, 171

Ulm, 171

Vahlen, 115Valentiner, 168van Dalen, 136van der Waall, 98van der Waerden, 21, 27, 47, 72, 96,

117–122, 125, 129, 142, 143,152, 166, 167, 228, 230, 231,233, 236

Vandiver, 71, 81, 83–85Veblen, 123, 145, 147, 150, 153von Kármán, 136

von Neumann, 72, 119, 162Vorbeck, 171

Wadsworth, 33Wang, 26, 50Weber, 228, 229Wedderburn, 40, 42, 47, 57, 59Weierstrass, 33, 114Weil, 112, 144, 224, 225Weissauer, 186Weizsäcker, 121Weyl, 71, 72, 117, 119–123, 127,

129–162, 166, 170, 174Whaples, 25Wheeler, 122, 123, 126, 154, 158, 162,

174Wichmann, 75, 170, 171Wielandt, 97, 106Wiener, 158, 172, 173Witt, 30, 75, 95, 101, 106, 108, 143,

191–193, 195–197, 202,204–206, 212

Yamamura, 88Yoshida, 98

Zassenhaus, 91, 92, 101, 103–106, 114Zelinsky, 68Zelmanov, 107Zermelo, 235Zorn, 74, 75

Subject Index

Albert-footnote, 56, 57, 63, 65, 67anisotropic, 201Arf invariant, 189–222Arf–Kervair invariant, 215Artin’s Reciprocity Law, 11, 27, 74,

80, 100, 140, 171Artinian ring, 41ascending chain condition, 138

Bernoulli number, 82, 83, 112Betti group, 135Brauer group, 5, 11, 32, 35, 36, 43, 55,

80, 206Brauer’s theorems, 6, 13Brauer–Hasse–Noether Theorem, see

Main TheoremBryn Mawr, 2, 72, 117–120, 122, 123,

126–128, 130, 132, 134, 147,154–158, 172–174

Burnside problem, 106, 108

cancellation theorem, 206central algebra, 4class field theory, 5, 10–12, 25, 26, 45,

70, 76, 79, 80, 87, 90, 94,101, 148, 187, 195, 224

class number, 80, 85, 86, 88second factor, 81, 82, 85

Clifford algebra, 198, 199, 201, 202,204–206, 212

cohomology group, 18, 43, 45complex multiplication, 80concurrent relation, 180continuity, 181crossed product, 17, 201cyclic algebra, 4, 55, 56, 60, 79

Dedekind ring, 41, 119, 138derivative, 182

differential module, 177Dirichlet principle, 135

elliptic curve, 84, 188elliptic function field, 62, 80, 195embedding problem, 245–247enlargement, 181Erlanger program, 137exponent, 13, 51, 53exponent-index theorem, 30, 60

factor system, 16Frobenius automorphism, 32, 38function field, 62, 75, 112, 130, 135,

136, 188, 192, 210, 212

generic point, 230groupoid, 41, 152Grün’s theorems

first, 92second, 102

Grunwald’s theorem, 25, 26Grunwald–WangTheorem, 26, 31, 247

Hasse algebra, 192, 205Hasse diagram, 92Hasse invariant, 5, 22, 35Hasse’s diary, 30, 87Hasse’s norm symbol, 38Hasse–Arf Theorem, 190, 224Hensel’s Lemma, 33, 232Hilbert space, 132Hilbert’s 13th problem, 18Hilbert’s 17th problem, 186Hilbertian field, 186hyperbolic plane, 201hyperreal, 178, 179

index, 13, 46, 51, 53infinitesimal, 175–179, 182, 183

278 Subject Index

integration, 182isotropic, 192, 206

Langlands program, 225linkage, 209, 210, 213local class field theory, 38, 80Local-Global Principle, 3, 11, 32, 49,

51, 52, 66, 67, 74, 75, 79,139, 192, 246–248

Main Theorem, 2, 4, 51–53, 62, 75maximal order, 41minimal splitting field, 47, 211model theory, 178, 180, 183, 184, 186Modern Algebra, 118modular representations, 27monad, 178

Noether equations, 17Noetherian ring, 134, 138nonstandard, 177, 178, 181–186, 188Norm Theorem, 12, 15, 16, 55, 66, 75,

76

p-adic number, 187, 232p-algebra, 73, 201Principal Genus Theorem, 45profinite group, 187

quadratic form, 191, 192, 195, 196

quadratic space, 196quaternion algebra, 50, 193, 203, 205

regular prime, 81, 83regular quadratic form, 198relativity theory, 136, 137representation theory, 46, 47, 142, 155Riemann matrices, 73, 155

Schur index, see index275Siegel’s pessimism, 29, 168Siegel–Mahler Theorem, 187, 188skew congress, 53, 54splitting algebra, 11

Tate–Poitou duality, 247, 248transfer, 93, 94

u-invariant, 210

valuation, 31, 232valuation ring, 178Vandiver’s conjecture, 81, 111

Wedderburn’s Theorem, 11, 142Witt equivalence, 207Witt ring, 206Witt vectors, 101, 195, 196

Zorn’s Lemma, 74, 235

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