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Graduate Texts in Mathematics
Editorial Board S. Axler F.W. Gehring K . A . Ribet
BOOKS OF RELATED INTEREST BY SERGE LANG
Math Talks for Undergraduates 1999, I S B N 0-387-98749-5
Linear Algebra, Third Edition 1987, I S B N 0-387-96412-6
Undergraduate Algebra, Second Edition 1990, I S B N 0-387-97279-X
Undergraduate Analysis, Second Edition 1997, I S B N 0-387-94841-4
Complex Analysis, Third Edition 1993, ISBN 0-387-97886
Real and Functional Analysis, Third Edition 1993, ISBN 0-387-94001-4
Algebraic Number Theory, Second Edition 1994, ISBN 0-387-94225-4
OTHER BOOKS BY LANG PUBLISHED BY
SPRINGER-VERLAG
Introduction to Arakelov Theory • Riemann-Roch Algebra (with William Fulton) • Complex Multiplication • Introduction to Modular Forms • Modular Units (with Daniel Kubert) • Fundamentals of Diophantine Geometry • Elliptic Functions • Number Theory III • Survey of Diophantine Geometry • Fundamentals of Differential Geometry • Cyclotomic Fields I and II • SL 2(R) • Abelian Varieties • Introduction to Algebraic and Abelian Functions • Introduction to Diophantine Approximations • Elliptic Curves: Diophantine Analysis • Introduction to Linear Algebra • Calculus of Several Variables • First Course in Calculus • Basic Mathematics • Geometry: A High School Course (with Gene Murrow) • Math! Encounters with High School Students • The Beauty of Doing Mathematics • THE FILE • CHALLENGES
Serge Lang
Algebra Revised Third Edition
Springer
Serge Lang Department of Mathematics Yale University New Haven, CT 96520 USA
Editorial Board S. Axler Mathematics Department San Francisco State
University San Francisco, CA 94132 USA [email protected]
F.W. Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA [email protected].
umich.edu
K.A. Ribet Mathematics Department University of California,
Berkeley Berkeley, CA 94720-3840 USA [email protected]
Mathematics Subject Classification (2000): 13-01, 15-01, 16-01, 20-01
Library of Congress Cataloging-in-Publication Data Algebra /Serge Lang.—Rev. 3rd ed.
p. cm.—(Graduate texts in mathematics; 211) Includes bibliographical references and index. ISBN 978-1-4612-6551-1 ISBN 978-1-4613-0041-0 (eBook) DOI 10.1007/978-1-4613-0041-0 1. Algebra. I. Title. II. Series.
QA154.3.L3 2002
512—dc21 2001054916
ISBN 978-1-4612-6551-1
Printed on acid-free paper.
This title was previously published by Addison-Wesley, Reading, MA 1993. © 2002 Springer Science+Business Media New York Originally published by Springer Science+Business Media LLC in 2002 Softcover reprint of the hardcover 3rd edition 2002 A l l rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
9 8 (corrected printing 2005)
springer.com
FOREWORD
The present book is meant as a basic text for a one-year course in algebra ,at the graduate level .
A perspective on algebra
As I see it, the graduate course in algebra must primarily prepare studentsto handle the algebra which they will meet in all of mathematics: topology,partial differential equations, differential geometry , algebraic geometry, analysis ,and representation theory, not to speak of algebra itself and algebraic numbertheory with all its ramifications . Hence I have inserted throughout references topapers and books which have appeared during the last decades , to indicate someof the directions in which the algebraic foundations provided by this book areused ; I have accompanied these references with some motivating comments, toexplain how the topics of the present book fit into the mathematics that is tocome subsequently in various fields; and I have also mentioned some unsolvedproblems of mathematics in algebra and number theory . The abc conjecture isperhaps the most spectacular of these.
Often when such comments and examples occur out of the logical order,especially with examples from other branches of mathematics , of necessity someterms may not be defined , or may be defined only later in the book . I have triedto help the reader not only by making cross-references within the book, but alsoby referring to other books or papers which I mention explicitly .
I have also added a number of exercises . On the whole, I have tried to makethe exercises complement the examples, and to give them aesthetic appeal. Ihave tried to use the exercises also to drive readers toward variations and applications of the main text , as well as toward working out special cases, and asopenings toward applications beyond this book .
Organization
Unfortunately, a book must be projected in a totally ordered way on the pageaxis, but that's not the way mathematics "is" , so readers have to make choiceshow to reset certain topics in parallel for themselves, rather than in succession .
v
vi FOREWORD
I have inserted cross-references to help them do this , but different people willmake different choices at different times depending on different circumstances .
The book splits naturally into several parts. The first part introduces the basicnotions of algebra. After these basic notions, the book splits in two majordirections: the direction of algebraic equations including the Galois theory inPart II; and the direction of linear and multilinear algebra in Parts III and IV.There is some sporadic feedback between them , but their unification takes placeat the next level of mathematics, which is suggested, for instance, in §15 ofChapter VI. Indeed, the study of algebraic extensions of the rationals can becarried out from two points of view which are complementary and interrelated:representing the Galois group of the algebraic closure in groups of matrices (thelinear approach), and giving an explicit determination of the irrationalities generating algebraic extensions (the equations approach) . At the moment, representations in GL2 are at the center of attention from various quarters, and readerswill see GL2 appear several times throughout the book . For instance, I havefound it appropriate to add a section describing all irreducible characters ofGL2(F) when F is a finite field. Ultimately, GL2 will appear as the simplest buttypical case of groups of Lie types, occurring both in a differential context andover finite fields or more general arithmetic rings for arithmetic applications .
After almost a decade since the second edition, I find that the basic topicsof algebra have become stable, with one exception . I have added two sectionson elimination theory, complementing the existing section on the resultant.Algebraic geometry having progressed in many ways, it is now sometimes returning to older and harder problems, such as searching for the effective constructionof polynomials vanishing on certain algebraic sets, and the older eliminationprocedures of last century serve as an introduction to those problems .
Except for this addition, the main topics of the book are unchanged from thesecond edition, but I have tried to improve the book in several ways.
First, some topics have been reordered . I was informed by readers and reviewers of the tension existing between having a textbook usable for relatively inexperienced students, and a reference book where results could easily be found ina systematic arrangement. I have tried to reduce this tension by moving all thehomological algebra to a fourth part, and by integrating the commutative algebrawith the chapter on algebraic sets and elimination theory, thus giving an introduction to different points of view leading toward algebraic geometry.
The book as a text and a reference
In teaching the course, one might wish to push into the study of algebraicequations through Part II, or one may choose to go first into the linear algebraof Parts III and IV. One semester could be devoted to each, for instance. Thechapters have been so written as to allow maximal flexibility in this respect, andI have frequently committed the crime of lese-Bourbaki by repeating short arguments or definitions to make certain sections or chapters logically independentof each other.
FOREWORD vii
Grant ing the material which under no circumstances can be omitted from abasic course, there exist several options for leadin g the course in various directions. It is impossible to treat all of them with the same degree of thoroughness.The prec ise point at which one is willing to stop in any given direction willdepend on time , place , and mood . However , any book with the aims of thepresent one must include a choice of topic s, pushin g ahead-in deeper waters ,while stopping short of full involvement.
There can be no universal agreement on these matter s, not even between theauthor and himself. Thus the concrete deci sions as to what to include and whatnot to include are finally taken on ground s of general coherence and aestheticbalance . Anyone teaching the course will want to impress their own personalityon the material, and may push certain topics with more vigor than I have, at theexpense of others . Nothing in the present book is meant to inhibit this.
Unfortunately, the goal to present a fairly comprehensive perspective onalgebra required a substantial increa se in size from the first to the second edition ,and a moderate increase in this third edition . The se increases require somedecisions as to what to omit in a given course.
Many shortcuts can be taken in the presentation of the topics, whichadmits many variations. For instance, one can proceed into field theory andGalois theory immediately after giving the basic definit ions for groups, rings,fields, polynomials in one variable, and vector spaces. Since the Galois theorygives very quickly an impression of depth, this is very satisfactory in manyrespects.
It is appropriate here to recall my or iginal indebtedness to Artin, who firsttaught me algebra. The treatment of the basics of Galois theory is muchinfluenced by the pre sentation in his own monograph.
Audience and background
As I already stated in the forewords of previous edit ions, the present bookis meant for the graduate level, and I expect most of those coming to it to havehad suitable exposure to some algebra in an undergraduate course, or to haveappropriate mathematical maturity. I expect students taking a graduate courseto have had some exposure to vector space s, linear maps, matrices, and theywill no doubt have seen polynomials at the very least in calculus courses .
My books Undergraduate Algebra and Linear Algebra provide more thanenough background for a graduate course . Such elementary texts bring out inparallel the two basic aspects of algebra , and are organized differently from thepresent book , where both aspect s are deepened. Of course, some aspects of thelinear algebra in Part III of the present book are more "elementary" than someaspects of Part II, which deal s with Galoi s theory and the theory of polynomialequations in several variables. Because Part II has gone deeper into the studyof algebraic equations, of necessity the parallel linear algebra occurs only laterin the total ordering of the book . Readers should view both part s as runningsimultaneously .
viii FOREWORD
Unfortunately, the amount of algebra which one should ideally absorb duringthis first year in order to have a proper background (irrespective of the subjectin which one eventually specializes) exceeds the amount which can be coveredphysically by a lecturer during a one-year course. Hence more material must beincluded than can actually be handled in class. I find it essential to bring thismaterial to the attention of graduate students.
I hope that the various additions and changes make the book easier to use asa text. By these additions, I have tried to expand the general mathematicalperspective of the reader, insofar as algebra relates to other parts of mathematics.
Acknowledgements
I am indebted to many people who have contributed comments and criticismsfor the previous editions, but especially to Daniel Bump, Steven Krantz, andDiane Meuser, who provided extensive comments as editorial reviewers forAddison-Wesley . I found their comments very stimulating and valuable in preparing this third edition. I am much indebted to Barbara Holland for obtainingthese reviews when she was editor. I am also indebted to Karl Matsumoto whosupervised production under very trying circumstances. I thank the many people who have made suggestions and corrections, especially George Bergmanand students in his class, Chee-Whye Chin, Ki-Bong Nam, David Wasserman,Randy Scott, Thomas Shiple, Paul Vojta, Bjorn Poonen and his class, in particular Michael Manapat.
For the 2002 and beyond Springer printings
From now on, Algebra appears with Springer-Verlag, like the rest of mybooks. With this change, I considered the possibility of a new edition, but decided against it. I view the book as very stable . The only addition which Iwould make , if starting from scratch, would be some of the algebraic propertiesof SLn and GLn (over R or C), beyond the proof of simplicity in Chapter XIII.As things stood, I just inserted some exercises concerning some aspects whicheverybody should know . The material actually is now inserted in a new editionof Undergraduate Algebra, where it properly belongs. The algebra appears as asupporting tool for doing analysis on Lie groups, cf. for instance Jorgenson/Lang Spherical Inversion on SLn(R), Springer Verlag 2001.
I thank specifically Tom von Foerster, Ina Lindemann and Mark Spencerfor their editorial support at Springer, as well as Terry Kornak and BrianHowe who have taken care of production.
Serge LangNew Haven 2004
Logical Prerequisites
We assume that the reader is familiar with sets, and with the symbols n, U,~ , C, E. If A , B are sets , we use the symbol A C B to mean that A is containedin B but may be equal to B . Similarly for A ~ B .
If f :A -> B is a mapping of one set into another, we write
X 1---+ f( x)
to denote the effect of f on an element x of A. We distinguish between thearrows -> and 1---+. We denote by f(A) the set of all elementsf(x), with x E A.
Let f :A -> B be a mapping (also called a map). We say that f is injectiveif x # y implies f(x) # f(y) . We say f is surjective if given b e B there existsa E A such that f(a) = b. We say that f is bijective if it is both surjective andinjective .
A subset A of a set B is said to be proper ifA # B.Let f :A -> B be a map, and A' a subset of A. The restriction of f to A' is
a map of A' into B denoted by fIA '.If f :A -> Band 9 : B -> C are maps, then we have a composite map 9 0 f
such that (g 0 f)(x) = g(f(x» for all x E A.Letf: A -> B be a map, and B' a subset of B. Byf- 1(B') we mean the subset
of A consisting of all x E A such that f(x) E B'. We call it the inverse image ofB'. We call f(A) the image of f.
A diagram
C
is said to be commutative if g of = h. Similarly, a diagram
A~B
•j j.C---->D
'"ix
X LOGICAL PREREQUISITES
is sa id to be commutative if 9 0 f = lj; 0 sp , We deal sometimes with morecomplicated diagrams, consisting of a rr ows between vario us objects. Suchdiagram s are ca lled commutati ve if, whenever it is poss ible to go from oneobject to another by means of two sequences of arrow s, say
A II A h I . -I A1~ 2~ " '~n
and
A 91 B 92 9 m -1 B Al~ 2~ " '~ m= n '
thenI n-l 0 • •• 0 II = 9m- 1 0 • •• °9b
in other words, the composite maps are equal. Most of our diagrams arecomposed of triangles or squares as above, and to verify that a diagram consisting of triangles or squares is commutative, it suffices to verify that eachtriangle and square in it is commutative.
We assume that the reader is acquainted with the integers and rationalnumbers, denoted respectively by Z and Q. For many of our examples, we alsoassume that the reader knows the real and complex numbers, denoted by Rand C.
Let A and I be two sets . By a family of elements of A, indexed by I, onemeans a map f: I-A. Thus for each i E I we are given an element f (i) E A .Alth ough a family does not differ from a map, we think of it as determining acollection of objects from A, and write it often as
or
writing a, instead of f(i) . We call I the indexing set.We assume that the reader knows what an equivalence relation is. Let A
be a set with an equivalence relation , let E be an equ ivalence class of elementsof A. We sometimes try to define a map of the equivalence classes into someset B. To define such a map f on the class E, we sometimes first give its valueon an element x E E (called a representative of E), and then show that it isindependent of the choice of representative x E E. In that case we say that fis well defined .
We have products of sets , say finite products A x B, or A I X .. . x An' andproducts of families of sets .
We shall use Zorn's lemma, which we describe in Appendix 2.We let # (S) denote the number of elements of a set S, also called the
cardinality of S. The notation is usually employed when S is finite. We alsowrite # (S) = card (S).
CONTENTS
Part One The Basic Objects of Algebra
3
3642
25
49
13
Groups3
Chapter I1. Monoids2. Groups 73. Normal subgroups4. Cyclic groups 235. Operations of a group on a set6. Sylow subgroups 337. Direct sums and free abelian groups8. Finitely generated abelian groups9. The dual group 46
10. Inverse limit and completion11. Categories and functors 5312. Free groups 66
Chapter II RingsI . Rings and homomorphisms 832. Commutative rings 923. Polynomials and group rings 974. Localization 1075. Principal and factorial rings 111
83
Chapter III Modules1. Basic definitions 1172. The group of homomorphisms 1223. Direct products and sums of modules 1274. Free modules 1355. Vector spaces 1396. The dual space and dual module 1427. Modules over principal rings 1468. Euler-Poincare maps 1559. The snake lemma 157
10. Direct and inverse limits 159
117
xi
xii CONTENTS
Chapter IV Polynomials1. Basic properties for polynomials in one variable2. Polynomials over a factorial ring 1803. Criteria for irreducibility 1834. Hilbert' s theorem 1865. Partial fractions 1876. Symmetric polynomials 1907. Mason-Stothers theorem and the abc conjecture8. The resultant 1999. Power series 205
173
194
173
Part Two Algebraic Equations
Chapter V Algebraic Extensions1. Finite and algebraic extensions 2252. Algebraic closure 2293. Splitting fields and normal extensions 2364. Separable extensions 2395. Finite fields 2446. Inseparable extensions 247
Chapter VI Galois Theory1. Galois extensions 2612. Examples and applications 2693. Roots of unity 2764. Linear independence of characters 2825. The norm and trace 2846. Cyclic extensions 2887. Solvable and radical extensions 2918. Abelian Kummer theory 2939. The equation X" - a = 0 297
10. Galois cohomology 30211. Non-abelian Kummer extensions 30412. Algebraic independence of homomorphisms13. The normal basis theorem 31214. Infinite Galois extensions 31315. The modular connection 315
Chapter VII Extensions of Rings1. Integral ring extensions 3332. Integral Galois extensions 3403. Extension of homomorphisms 346
308
223
261
333
CONTENTS xiii
Chapter VIII Transcendental Extensions 355I. Transcendence bases 3552. Noether normalization theorem 3573. Linearly disjoint extensions 3604 . Separable and regular extensions 3635. Derivations 368
Chapter IX Algebraic SpacesI. Hilbert' s Nullstellensatz 3782. Algebraic sets , spaces and varieties3. Projections and elimination 3884. Resultant systems 4015. Spec of a ring 405
381
377
Chapter X Noetherian Rings and ModulesI . Basic criteria 4132. Associated primes 4163. Primary decomposition 4214. Nakayama's lemma 4245. Filtered and graded modules 4266. The Hilbert polynomial 4317. Indecomposable modules 439
413
Chapter XI Real FieldsI . Ordered fields 4492. Real fields 4513. Real zeros and homomorphisms 457
449
Chapter XII Absolute ValuesI. Definitions , dependence , and independence 4652. Completions 4683. Finite exten sions 4764. Valuations 4805. Completions and valuations 4866. Discrete valuations 4877. Zero s of polynomials in complete fields 491
465
Part Three Linear Algebra and Representations
Chapter XIII Matrices and Linear Maps1. Matrices 5032. The rank of a matrix 506
503
xiv CONTENTS
3. Matrices and linear maps 5074. Determinants 5115. Duality 5226. Matrices and bilinear forms 5277. Sesquilinear duality 5318. The simplicity of SL2(F)/± I 5369. The group SLn(F), n 2: 3 540
Chapter XIV Representation of One Endomorphism 5531. Representations 5532. Decomposition over one endomorphism 5563. The characteristic polynomial 561
Chapter XV Structure of Bilinear Forms 5711. Preliminaries, orthogonal sums 5712. Quadratic maps 5743. Symmetric forms, orthogonal bases 5754. Symmetric forms over ordered fields 5775. Hermitian forms 5796. The spectral theorem (hermitian case) 5817. The spectral theorem (symmetric case) 5848. Alternating forms 5869. The Pfaffian 588
10. Witt's theorem 58911. The Witt group 594
Chapter XVI The Tensor Product 6011. Tensor product 6012. Basic properties 6073. Flat modules 6124. Extension of the base 6235. Some functorial isomorphisms 6256. Tensor product of algebras 6297. The tensor algebra of a module 6328. Symmetric products 635
Chapter XVII Semisimplicity 6411. Matrices and linear maps over non-commutative rings 6412. Conditions defining semisimplicity 6453. The density theorem 6464. Semisimple rings 6515. Simple rings 6546. The Jacobson radical, base change, and tensor products 6577. Balanced modules 660
CONTENTS XV
Chapter XVIII Representations of Finite Groups 6631. Repre sentations and semisimplicity 6632. Characters 6673. I-d imensional repre sentations 6714. The space of clas s function s 6735. Orthogonality relations 6776. Induced characters 6867. Induced repre sentations 6888. Positive decomposition of the regular character 6999. Supersolvable groups 702
10. Brauer's theorem 70411. Field of definition of a representation 71012. Example : GL2 over a finite field 712
Chapter XIX The Alternating Product1. Definition and basic properties 7312. Fitting ideals 7383. Universal derivations and the de Rham complex4 . The Clifford algebra 749
746
731
Part Four Homological Algebra
Chapter XX General Homology Theory1. Complexes 7612. Homology sequence 7673. Euler characteristic and the Grothendieck group 7694 . Injective modules 7825. Homotopies of morphisms of complexes 7876. Derived functors 7907. Delta-functors 7998. Bifunctors 8069. Spectral sequences 814
Chapter XXI Finite Free Resolutions1. Special complexes 8352. Finite free resolutions 8393. Unimodular polynomial vectors 8464. The Koszul complex 850
761
835
Appendix 1Appendix 2BibliographyIndex
The Transcendence of e and Tt
Some Set Theory867875895903
