Selected Titles in This Series 48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole),

Rings, modules, and algebras in stable homotopy theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of

associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental groups of

compact Kahler manifolds, 1996 43 James E. Humphreys, Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Preese, Jaroslav Jezek, and J. B. Nation, Free lattices, 1995 41 Hal L. Smith, Monotone dynamical systems: an introduction to the theory of

competitive and cooperative systems, 1995 40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite

simple groups, number 2, 1995 40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite

simple groups, number 1, 1994 39 Sigurdur Helgason, Geometric analysis on symmetric spaces, 1993 38 Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, 1993 37 Leonard Lewin, Editor, Structural properties of polylogarithms, 1991 36 John B. Conway, The theory of subnormal operators, 1991 35 Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, 1990 34 Victor Isakov, Inverse source problems, 1990 33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean

fields, 1990 32 Howard Jacobowitz, An introduction to CR structures, 1990 31 Paul J. Sally, Jr. and David A. Vogan, Jr., Editors, Representation theory and harmonic

analysis on semisimple Lie groups, 1989 30 Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, 1989 29 Alan L. T. Paterson, Amenability, 1988 28 Richard Beals, Percy Deift, and Carlos Tomei, Direct and inverse scattering on the line,

1988 27 Nathan J. Fine, Basic hypergeometric series and applications, 1988 26 Hari Bercovici, Operator theory and arithmetic in H°°, 1988 25 Jack K. Hale, Asymptotic behavior of dissipative systems, 1988 24 Lance W. Small, Editor, Noetherian rings and their applications, 1987 23 E. H. Rothe, Introduction to various aspects of degree theory in Banach spaces, 1986 22 Michael E. Taylor, Noncommutative harmonic analysis, 1986 21 Albert Baernstein, David Drasin, Peter Duren, and Albert Marden, Editors, The

Bieberbach conjecture: Proceedings of the symposium on the occasion of the proof, 1986 20 Kenneth R. Goodearl, Partially ordered abelian groups with interpolation, 1986 19 Gregory V. Chudnovsky, Contributions to the theory of transcendental numbers, 1984 18 Frank B. Knight, Essentials of Brownian motion and diffusion, 1981 17 Le Baron O. Ferguson, Approximation by polynomials with integral coefficients, 1980 16 O. Timothy O'Meara, Symplectic groups, 1978 15 J. Diestel and J. J. Uhl, Jr., Vector measures, 1977 14 V. Guillemin and S. Sternberg, Geometric asymptotics, 1977 13 C. Pearcy, Editor, Topics in operator theory, 1974

(Continued in the back of this publication)

http://dx.doi.org/10.1090/surv/048

Integer-Valued Polynomials

Mathematical Surveys

and Monographs

Volume 48

Integer-Valued Polynomials

Paul-Jean Cahen Jean-Luc Chabert

American Mathematical Society

Editorial Board Georgia M. Benkart Tudor Stefan Ratiu, Chair Howard A. Masur Michael Renardy

The first author was supported in part by the CNRS URA 225, and the second author by the LaMfa-Amiens.

1991 Mathematics Subject Classification. Primary 13A15, 13G05; Secondary 11R04, 13B25, 13C20, 13E99, 13F05, 13F20.

Library of Congress Cataloging-in-Publication D a t a Cahen, Paul-Jean, 1946-

Integer-valued polynomials / Paul-Jean Cahen, Jean-Luc Chabert. p. cm.—(Mathematical surveys and monographs, ISSN 0076-5376 ; v. 48)

Includes bibliographical references and index. ISBN 0-8218-0388-3 (alk. paper) 1. Polynomials. 2. Ideals (Algebra) 3. Rings of integers. I. Chabert, Jean-Luc. II. Title.

III. Series: Mathematical surveys and monographs ; no. 48. QA161.P59C34 1996 512'.4—dc20 96-35954

CIP

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionQams.org.

© 1997 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

10 9 8 7 6 5 4 3 2 1 02 01 00 99 98 97

Contents

Foreword xi

Historical Introduction xiii

Mathematical Introduction xvii

Conventions and Notation xix

Chapter I. Coefficients and Values 1 1.1. Various rings of integer-valued polynomials 2 1.2. Localization 6 1.3. Trivial cases 9 1.4. Polynomial torsion theory 14 Exercises 19

Chapter II. Additive Structure 25 II. 1. Characteristic ideals 26 11.2. Discrete valuation domains 29 11.3. Dedekind domains 34 11.4. Polya fields 40 Exercises 43

51 52 53 56 60 63 66

Chapter III.l. III.2. III.3. III.4. III.5.

III. Stone-Weierstrass a-adic topology Continuity Mahler series Reduction to one-dimensional local domains Analytically irreducible domains

Exercises

viii CONTENTS

Chapter IV. Integer-Valued Polynomials on a Subset 73 IV. 1. Polynomial closure 74 IV.2. Globalization 79 IV.3. Dedekind domains as polynomially dense subrings 82 IV.4. Integral closure 86 Exercises 90

Chapter V. Prime Ideals 99 V.l. Fibers and Krull dimension 100 V.2. Discrete valuation domains 103 V.3. Unibranched and non-unibranched domains 108 V.4. Large fibers 111 Exercises 114

Chapter VI. Multiplicative Properties 123 VI. 1. Priifer domains and valuation overrings 124 VI.2. Various transfer properties 129 VI.3. Factorization properties 134 VI.4. Priifer or not Priifer 140 Exercises 151

Chapter VII. Skolem Properties 159 VII. 1. Skolem properties of rings of algebraic integers 160 VII.2. Unitary and non-unitary ideals 165 VII.3. Strong Skolem properties 170 VII.4. Almost Skolem rings 175 VII.5. Skolem rings and the Nullstellensatz property 178 Exercises 183

Chapter VIII. Invertible Ideals and the Picard Group 193 VIII.l. The short exact sequence 194 VIII.2. Invertible unitary ideals and value functions 198 VIII.3. Generalization to analytically irreducible domains 203 VIII.4. The strong two-generator property 208 VIII.5. Prime ideals and the Picard group 213 Exercises 220

Chapter IX. Integer-Valued Derivatives and Finite Differences 227 IX. 1. More rings of integer-valued polynomials 228 IX.2. Prime ideals 233 IX.3. Bases for finite differences 241 IX.4. Bases for derivatives 247 Exercises 252

CONTENTS ix

Chapter X. Integer-Valued Rational Functions 257 X.l. From polynomials to rational functions 258 X.2. Continuity and prime ideals 261 X.3. Multiplicative properties 265 X.4. Valuation domains revisited 270 Exercises 277

Chapter XL Integer-Valued Polynomials in Several Indeterminates 285 XI. 1. From one to several indeterminates 286 XL 2. Prime ideals and related topics 291 XI.3. The Nullstellensatz property 295 XI.4. Miscellaneous 301 Exercises 303

References 307

List of Symbols 317

Index 319

Foreword

Integer-valued polynomials on the ring of integers have been known for a long time and used in calculus. Polya and Ostrowski generalized this notion to rings of integers of number fields. More generally still, one may consider a domain D and the polynomials (with coefficients in its quotient field) mapping D into itself. They form a D-algebra — that is, a D-module with a ring structure. Appearing in a very natural fashion, this ring possesses quite a rich structure and the very numerous questions it raises allow a thorough exploration of commutative algebra.

Here is the first book entirely devoted to this topic. The idea of writing such a monograph arose at a meeting on Integer-valued Polynomials held in Decem­ber 1990 at the C.I.R.M. (Centre International de Rencontres Mathematiques), Marseilles, Prance. The project was launched and took shape while we taught an intensive course on this topic for advanced students at La Sapienza in Roma, upon the invitation of Marco Fontana in April and May 1993. The book was finished while the first author (in alphabetical order) was in the U.S.A., first at Charlotte (NC), where he was invited by Evan Houston, and then at Tallahassee (FL), upon the invitation of Robert Gilmer. In Florida the second author came for a short but very profitable visit, and a graduate course was also offered; this proved to be quite useful in the editing process.

We thank all those here named, and many others, for their support. Espe­cially, we thank Robert Gilmer for his very numerous and very useful comments.

May 1996 Paul-Jean Cahen and Jean-Luc Chabert Tallahassee, Paris

Historical Introduction

For all positive integers x and n, the binomial coefficient (*) is clearly an integer, hence the polynomial (*) = —-——~j—~n+ is integer-valued, although its coefficients are not in Z. In fact, it seems to have been known from quite an early date that every integer-valued polynomial is a linear combination, with coefficients in Z, of the polynomials ( n ) . Indeed, they appeared as early as the seventeenth century in interpolation formulae: linear interpolation failed sometimes to provide accurate approximations, and Isaac Newton was one of the first mathematicians to use finite differences to tabulate square and cubic roots. The corresponding formula was given for the first time by James Gregory in a letter to John Collins dated 23 November 1670:

"... / remember ye did once desire of me my methode of finding the propor­tional parte in tables, which is this: In figura octava mearum exercitationum, in recta AI imaginetur qucelibet Aa, cui sit perpendicularis cry, sitque 7 in curva ABH, reliquis stantibus ut prius, sit series infinita f ) ^ ) i 7 ) i 7 , etc., fiatque productum ex duobus primis seriei terminis | , ex tribus primis k, ex quatuor primis -c, ex quinque primis ™, etc., in infinitum; erit recta a 7 — l f + c ~ ' ~ l ? 1 + ^ + e^c-> ^n infinitum-' this methode, as I apprehend, is both more easie and universal then ether Briges or Mercators, and also performed without tables..." [in The correspondence of Isaac Newton, vol. I (1661-1675), p. 46, edited by H.W. Turnbull, Cambridge University Press, 1959]

I K I , M N O P

Taken from James Gregory Exercitationes Geometricae (figure 8), London, 1668

xiii

XIV HISTORICAL INTRODUCTION

Writing it in current terms we obtain

f{hx)« £>*/(o) * (*"1 ) (*~2^- ( x" f e + 1 )

k=o k'

where the finite differences Afc/(0) are defined as follows:

A/(0) = f(h) - /(0) and Afc/(0) = A(A f e"1 /(0)) .

It expresses the interpolation of a function / at equidistant points 0, ft,..., nft. This formula is similar to the Taylor-MacLaurin expansion and is currently

known as the Gregory-Newton interpolation formula. However, as early as the beginning of the seventeenth century, Thomas Harriot had developed it with n equal to 5, and Henry Briggs had used it to tabulate his logarithms.

In particular, if / is a polynomial with degree n and ft is taken as unit, then:

/(*) = ±AV(0) *(*-l)(«-2)-('-* + l) = £A*/(0)(f), k=0 * k=0 ^ '

where k , , v

A/(0) = / ( l ) - / ( 0 ) and A*/(0) = A(A f e"1/(0)) = £ ( - l ) f c - M . ) / ( t ) . i=o W

It is clear that if the polynomial / takes integral values at integers, then the expressions A /c/(0) are integers, and conversely, since the polynomials (*) are integer-valued.

Nevertheless, the polynomials (*) were then only tools to compute tables. It was only in 1919 that two papers by Georg Polya and Alexander Ostrowski, both titled "Uber ganzwertige Polynome in algebraischen Zahlkorpern" (On integer-valued polynomials in algebraic number fields), considered integer-valued poly­nomials in their own right, seemingly following suggestions of A. Hurwitz. (One may point out that Polya was actually interested in entire functions that take integral values on the integers.)

Polya and Ostrowski considered the integer-valued polynomials on an algebraic number field K, that is, the polynomials f(X), with coefficients in K, which take integral values on integers of K (for instance, the polynomial \ + ~ ^ , with coefficients in the quadratic field Q[z], is such that its value on every Gaussian integer, that is, on every element of the ring Z[i], is a Gaussian integer).

Clearly the set of integer-valued polynomials in K is a module over the ring of integers of K, and one may ask if it has a basis as a D-module, as it occurs for K — Q. More precisely, Polya raised the question of whether there is regular basis, that is, a sequence of integer-valued polynomials {/&} such that deg(/^) = k and each integer-valued polynomial with degree n may be written in the form:

/ = ao/o + a i^1 "• *" anfn

HISTORICAL INTRODUCTION xv

with coefficients ak in the ring of integers of K. Polya and Ostrowski obtained some nice results. For example, there exists a regular basis if and only if the products of prime ideals of every given norm are principal ideals. In fact, one could trace this idea back to an 1896 paper by Kurt Hensel. He dealt there with the greatest common denominator of a set of integers that can be represented by a polynomial, in other words, that are the values of an integer-valued polynomial. He introduced in particular a fundamental set of polynomials which anticipated Polya's notion of a regular basis.

The most significant studies following Polya and Ostrowski's papers con­cerned only rational integers. First, in 1936, Thoralf Skolem considered the set of integer-valued polynomials in Q not only as a Z-module, but as a ring; he proved a version of the Bezout identity that is not true in Z[X]. If f \ , . . . , fm are integer-valued polynomials such that, for each integer a, the greatest common di­visor of / i ( a ) , . . . , / m (a ) is equal to 1, then there are integer-valued polynomials 9u-- i9m such that:

fl9l H h frnQm = 1. Next, in 1951, Ernst Straus considered the set of integer-valued polynomials

whose derivatives of all orders are also integer-valued. This is again a Z-module for which Straus described a basis: a polynomial / belongs to this module if and only if

k=o x 7

where the coefficients ak are integers and bk is given by the formula

h= n p[i]-p prime

In 1955, Nicolaas G. de Bruijn characterized in a similar way integer-valued polynomials / , all of whose finite divided differences of first order are also integer-valued: the coefficient bk is now replaced by the least common multiple of {1,2 , . . . ,&}. Then, in 1959, Leonard Carlitz extended De Bruijn's results to integer-valued polynomials whose finite divided differences of order i, 1 < i < n, are integer-valued. He proved that integer-valued polynomials whose finite di­vided differences of all orders are integer-valued are the same as integer-valued polynomials whose derivatives of all orders are integer-valued.

In this book we present various studies on integer-valued polynomials that have been developed in the last quarter century. Thus, we end this brief historical review and turn next to a short survey of the topics covered in this monograph, using the tools of commutative algebra.

xvi HISTORICAL INTRODUCTION

ttber ganzwertige Polynome in algebraischen Zahl-kOrpern.

Von Herrn Oeorg P6lya in Zurich.

1. Man nennt das Polynom

(1.) P(x) = a 0 a f - f ax2r-1 + . . . +am

ganzzahJig im Korper K, wenn die Koeffizienten a0, a^ ... am ganze Zahlen von K sind. Man nenne ein Polynom P(x) ganzwertig im Korper K, wenn fiir jede ganze Zahl f von K der Wert P ( | ) eine ganze Zahl von K wird. 1st das Polynom (1.) ganzwertig im Korper K, so miissen er-sichtlicherweise samtliche Koeffizienten or0, alt . . . am im Korper K liegen. Aber sie brauchen nicht alle ganze Zahlen zu sein, d. h. es gibt ganzwertige Polynome, die nicht ganzzahlig sind. So sind z. B. die Polynome

x(x- 1) *(x-l) (1 + ^ 5 ) J ( S - 1 ) 2 ' i + 1 ' 2

ganzwertige Polynome bzw. in den Korpern K(l), K(i)} K(Y— 5). Be-merken wir noch, dafi das ersterwahnte Polynom weder in K(i) noch in £ (]/__ 5) ganzwertig ist, da es fiir x = i bezw. fiir x — 1 -f V— 5 keinen ganzzahligen Wert annimmt.

Es gibt Korper K. in denen eine unendliche Folge von ganzwer-tigen Polynomen

Journal far Sfatheautik. Bd. 149. Heft 3/4. 13

I

Introduction of Polya's paper: On integer-valued polynomials in algebraic number fields in

Journal fiir die reine und angewandte mathematik, 149 (1919), p. 97

Reprinted by permission from Walter de Gruyter & Co., Berlin and New York.

Mathematical Introduction

Let D b e a domain with quotient field K and let Int(D) be the set of integer-valued polynomials on D, that is,

Int(£>) = {/ G K[X] | /(£>) C D}.

Clearly lnt(D) is a D-module, it is a subring of If [X], and it contains £>[X]. On one hand we will describe bases for the D-module Int(D) as Polya and Ostrowski did for rings of integers. On the other hand, we will study properties of the ring Int(D) in connection with those of the ring D. This latter point of view seems to be relatively new and interesting. For example, it is worth noticing that the ring Int(Z) is not Noetherian and that it is probably the "most natural" example of a non-Noetherian ring that does not result from an ad hoc construction.

To study Int(D), the first tool we consider is localization. As it turns out (not so obviously), integer-valued polynomials behave well with respect to local­ization: if / is an integer-valued polynomial on D, and S is a multiplicative set, then / is integer-valued on the quotient ring S~1D of D with respect to S. In fact, if D is Noetherian, the rings S_ 1Int(D) and In t (S - 1 D) are equal.

We then raise a natural question: when does the equality Int(D) = D[X] hold? We shall easily see, using Cramer's rule, that this occurs if D is a local domain with infinite residue field. This is the reason why the (necessarily maximal) prime ideals with a finite residue field play a central role throughout this book.

As a matter of fact, we are mostly concerned with one-dimensional Noetherian domains D with finite residue fields, natural extensions of rings of algebraic integers. Moreover, we may localize, and thus suppose that D is local, with maximal ideal m.

Considering the m-adic topology, we prove that integer-valued polynomials are uniformly continuous functions from D to D (where D denotes the completion of D). We even extend Mahler's results on p-adic continuous functions, proving an m-adic version of the Stone-Weierstrass approximation theorem:

XVI1

xviii MATHEMATICAL INTRODUCTION

Int(D) is dense, for the uniform convergence topology, in the ring of continuous functions C(D,D) if and only if D is analytically irreducible (that is, D is an integral domain).

These analytic results shed light on the structure of Int(D). For instance, the prime ideals lying over the maximal ideal m of D are of the form

9Km,a = {/ € Int(D) | f(a) e mD}

where a is an element of D. Moreover, these ideals are in one-to-one correspon­dence with the elements of D when D is analytically irreducible.

This latter assumption is one of the restrictive hypotheses we are going to consider. Sometimes, we will need only a slightly weaker assumption: we let D be unibranched, that is, its integral closure is a local ring. And throughout, we encounter one or the other of these two hypotheses. Nevertheless, most results are simpler if we even restrict ourselves to a rank-one discrete valuation domain. We often deal with this particular situation first, and then determine under which hypotheses our results are still valid.

Although Int(Z) is not Noetherian, thus not a Dedekind domain, it is a two-dimensional Priifer domain. If, in particular, Zp is the ring of p-adic integers, then Int(Zp) provides a nice counterexample: it is a completely integrally closed two-dimensional Priifer domain that is not an intersection of rank-one valuation domains. More generally, we characterize the domains D (among which are the rings of integers of a finite algebraic number field) such that lnt(D) is a Priifer domain.

In a similar vein as the property that Skolem established for Int(Z), we also study the relation between an ideal / of Int(Z)) and its ideals of values 1(a) = {f(a) I / € / } . Among many other results, we obtain an integral version of the Hilbert Nullstellensatz: if I is finitely generated and if g is a polynomial such that g(a) belongs to 1(a) for each a in JD, then g belongs to the radical of / .

Recall that the Picard group of a domain is the class group of its invertible fractional ideals. We study the invertible ideals of Int(D). In the local case (when D is unibranched) we show that the classes of the finitely generated prime ideals generate the Picard group.

We also extend the studies of De Bruijn, Straus and Carlitz concerning deriva­tives and finite divided differences.

The studies alluded to before may also be considered for integer-valued ra­tional functions. This is essentially done in the local case, since otherwise an integer-valued rational function is frequently a polynomial.

Finally, we shall consider integer-valued polynomials and rational functions in several indeterminates.

Conventions and Notation

R denotes a commutative ring with unity element. D denotes an infinite integral domain with quotient field K. S~lD denotes the quotient ring of D with respect to a multiplicative subset 5. V denotes a valuation domain.

Z(p) is the localization of Z with respect to the prime ideal (p). Zp is the ring of p-adic integers. ¥q is the finite field with q elements. P denotes the set of prime numbers.

Ac B means AC B and A ^ B. R is an overring of D means D C R C K.

If not specified, — a local ring is not necessarily Noetherian, — an ideal is an integral ideal, — a discrete valuation is a rank-one discrete valuation, — almost all means all but finitely many.

At the end of the book, the reader will find an index, with, in regard of each entry, the number of the page where the corresponding term is defined. For the sake of completeness, some classical definitions are recalled in the course of the text; they are recorded in the index.

XIX

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[AM] M. F. Atiyah and I.G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.

[BS] M. F. Borevitch and I. R. Shafarevich, Teoria Cisel, Moscow, 1964 (Rus­sian); English transl. Number Theory Academic Press, New York, 1966.

[B] N. Bourbaki, Algebre Commutative, Hermann, Paris, 1961-1965 (French); English transl. Addison-Wesley, Readings, 1972.

[E] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geome­try, Springer-Verlag, New York, 1995.

[G] R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972; rep. Queen's Papers in Pure and Applied Mathematics, vol. 90, Queen's Uni­versity, Kingston, 1992.

[K] I. Kaplansky, Commutative Rings, University of Chicago Press, Chicago and London, 1974.

[L] J. Lambek, Torsion Theories, Additive Semantics and Rings of Quotients, Lecture Notes, vol. 177, Springer-Verlag, Berlin, Heidelberg, New York, 1971.

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List of Symbols

O , *ii Z[i] , xiv ¥q , xix Int(Z) , 2 (D :D x) , 11 ann(a;) , 17 3n{B) , 26 MO > 29 / n , 3 2 rn , oo *% , 34 Max(£>) , 35 «(£>) , 37 "q,n i o\)

-Dx ,41 £] . 46 C ( F , Q p ) , 5 1 # , 5 2 E ,65 C(#, fl) , 69 (D : R) , 87 Cm(n) , 93 Fg ,96 dim(D) , 101 9Km,a . 103 a»C,a, no va , 127 Int(Zp) , 130 £fl(s) , 134 $(R,n) , 135

A / , xiv Z(p) , xix P , xix Int(D) , 3 (f(R)) , 13 Ass(M) , 17 3n , 26 [x] , 30 9*n , 32 v(g(V)) , 34 AT(m) , 34 Int„(£») , 36 Uq ,37 Q( y/l) , 40 Hl{G,U) ,42 cqtuU*), 51 C(ZP,ZP) ,51 a , 52 C(£, 5) , 65 £ , 74 Up ,91 im(n) , 93 V,, ioo dim„(D) , 102 9ttp,a , 106 / # , 109 Va , 127 £ , 133 /£>(£) , 135 vm\K* , 140

A f c/ , xiv Zp , xix

( * ) . 2 Int (£,£>) , 3 C( / ) , 13 Supp(M) , 18 3-1 , 28 w9(n) , 30 Fq ,33 «($) , 34 wm , 34 C{D) , 37 7Tg , 38

Q(V3) , 41 card(£) , 43 Q P J 51 c(5,5), 51 C(V,V),57 C(V,k) ,68 E°,79 C(n) , 93 am(n) , 94 ¥p,a , 100 Int(E, a) , 102 £ , 110 qm , 109 Fp((T)) , 130 IR(X) , 134 p(R) , 135 em{K/Kt) , 140

317

318 LIST OF SYMBOLS

fm(K/K.) , 140 [K : K.] , 148 21* , 161 X(R) , 195 U(R),195 Pic(iJ) , 195 C{D,X) , 199 Wi , 200 C(V, Z) , 200 K0(lnt(D)) , 213 % , 218 hd,W(E,D) ,228 Ahf(X) , 228 Int[°°](D) , 229 TC{kJ , 233 aR£2,237 dfe(n) , 242 In , 243 L(n] , 245 Afc(n) , 246 Int(fc)(Z) , 249 IntR(Z) , 258 ?K,a , 265 SB , 271 q 3 | , 272 Va.n , 274 V* , 274 a«!U, 277 Int(£, D) , 286 Int(£>n) , 286 d e g X i ( / ) , 2 9 0 C{Dn,D) ,292 9Jtm,s , 294 Da , 295 Eh , 301 J O ® , 301 IntW(£>n) ,305

#(«•) , 140 +D , 155 NL/K , 168 P(i?) , 195 Fu(lnt(D)) , 195 Pic(lnt(D)) , 195 wf , 199 C(D,X(D)) , 200 C(V,Z)/Z,202 [2t] , 216 % , 219 Int^iD) , 228 A h l , . . . , h f c / (X),229 Int*(£>) , 229 Is.m > 235 V(D/Z) , 240 «*(n) , 242 Int[fc)(Z) , 244 Int[°°](Z) , 245 Aoo(n) , 246 Int<°°)(Z) , 251 IntR (E,D) ,259 %% , 265 w ,271 E' , 272 V»«,n , 274 5DT , 274 7 P 0 , 278 X ,286 21(a) , 287 Z4 , 291 C(Dn,D) ,293 £ , 2 9 5 J(JJ) , 296 Iht<*>(Dn) , 301 IntR (E,D) ,302

F(v.) , 140 21(a) , 160 F(R) , 194 F* , 195 J u( lnt(D)) , 195 % > 198 C(D,T(D)) , 199 w , 200 Pic(lnt(Z)) , 203 PPJ - 216 Int<*> (£?,£>) ,228 A fc/(a) , 228 IntW(£>) ,229 - " n n , a 5 ^ * 3

an£!a , 236 aB(n) ,241 I n t ^ Z ) ,243 ln

k) , 244 Inttfc](Fq[i]) ,246 IntW(Z) , 249 IntR(£>) , 258 TE , 264 *Pp,a , 268 K' ,271 SC,W > 273 u* , 274 9^o,o , 276 £ , 2 8 6 a , 286 Int(Z") , 290 ¥p,a , 292 1 , 294 »C,a . 295 3 D ( # ) , 296 Int(2o) p n ) ^ 3 0 1

IntR(D") , 302

Index

ACCP, 133 adic topology, 52 algebraic order, 169 almost

all, xix Dedekind domain, 125 integer-valued, 3 integral, 95, 129 Skolem, 169, 295 strong Skolem, 169

analytically irreducible, 63 locally, 173

annihilator, 17 arithmetically dense, 81 Artin-Rees lemma, 54 associated prime, 12 atom, 134 atomic domain, 134

Bezout domain, 127 basis

binomial, 2 Fermat, 38 regular, 27

BFD, 135 binomial

basis, 2 coefficient, xiii Fermat, 33, 38 Gaussian, 46 polynomial, 2

bounded factorization domain, 135

cancellation property, 187 characteristic

ideal, 26, 291 sequence, 241

class group, 37, 195 value functions, 201

clopen subset, 52 closure

polynomial, 74 Skolem, 161, 295

coherent ring, 132 common point modulo TTln, 275 completely integrally closed, 129 completion, 52, 67 conductor, 11, 87 content module, 13

Dedekind domain, 34 dense

arithmetically, 81 poly normally, 74, 287

Dirichlet's theorem, 91 divided ideal, 10 divisibility group, 195 divisorial ideal, 75 domain

almost Dedekind, 125 analytically irreducible, 63 atomic, 134 Bezout, 127 bounded factorization, 135 completely integrally closed, 129 Dedekind, 34 discrete valuation, 13 half-factorial, 135 Jaffard, 102 Krull, 8 locally analytically irreducible, 173 locally unibranched, 214 Mori, 139 one-dimensional, 11 Priifer, 124 pseudo-valuation, 119 semi-local, 210 semi-normal, 154 unibranched, 63 unique factorization, 134 valuation, 13

double boundedness condition, 140 d-ring, 165

319

320 INDEX

elasticity, 135 equalizing ideal, 110, 235 essential

extension, 15 valuation, 8, 124

extension essential, 15 immediate, 78 radical, 88

Fermat basis, 38 binomial, 33, 38 polynomial, 33, 39

field local, 130 perfect, 182 prime, 145

filter, 119 filtration, 66 finite difference, 228 fractional

ideal, 26 subset, 4, 286

function integer-valued, 5

rational, 258 periodic modulo m, 57 uniformly locally constant, 199 value, 198

function field, 42 fundamental unit, 41

Gaussian binomial coefficient, 46 integer, xiv

G-domain, 188 G-ideal, 176 Goldman ideal, 176 good behavior under localization, 126 Gregory-Newton formula, xiv group

Cartier divisors, 195 class, 37, 195 divisibility, 195 invertible ideals, 195 invertible unitary ideals, 195 Polya-Ostrowski, 37 Picard, 195

half-factorial domain, 135 Hasse existence theorem, 148 height, 100 height-one prime, 8 Hilbert

Nullstellensatz, 181

property, 169 ring, 176 strong property, 169

ideal annihilator, 17 associated prime, 12 characteristic, 26, 291 conductor, 11 divided, 10 divisorial, 75 equalizing, 110, 235 fractional, 26 Goldman, 176 inverse, 28 invertible, 28 locally principal, 208 n-generated, 208 norm, 34 pointed maximal, 273 pseudo-principal, 11 Skolem closed, 161, 295 strongly n-generated, 208 totally decomposed, 85 unitary, 124 values, 160, 198, 267, 287

immediate extension, 78 subextension property, 145 valuation, 145

integer-valued function, 5

entire, 5 rational, 258

polynomial, 3 invertible ideal, 28

Jacobson radical, 76 ring, 177

Jaffard domain, 102

Krull dimension, 100 domain, 8

Legendre formula, 30 local field, 130 locally

analytically irreducible, 173 principal, 208 unibranched, 214 uniformly locally constant, 207

Mahler series, 56 module

INDEX 321

content, 13 support, 18 torsion, 15 torsion-free, 15 values, 13

Mori domain, 139 Mori-Nagata theorem, 88

n-generated, 208 strongly, 208

Noetherian space, 132 norm

element, 168 function, 55 ideal, 34

normalized valuation, 141 Nullstellensatz property, 180, 296 number field, 40

*-operation, 186 overring, xix

Polya field, 40 Polya-Ostrowski group, 37 perfect field, 182 period, 57 Picard group, 195 point at infinity, 271 pointed maximal ideal, 273 polynomial

almost integer-valued, 3 binomial, 2 closure, 74 Fermat, 33, 39 integer-valued

in several indeterminates, 286 on a domain, 3 on a subset, 3

torsion theory, 16 polynomially

closed, 74 dense, 74, 287 equivalent, 74

Priifer domain, 124 prime

associated, 12 decomposition, 40, 148 height, 100 height-one, 8 ramified, 40 totally decomposed, 82

prime field, 145 pseudo-

principal ideal, 11 valuation domain, 119

radical

extension, 88 Jacobson, 76

ramification index, 40, 140, 148 ramified prime, 40 rational function

integer valued on a domain, 258 integer-valued on a subset, 259

regular basis, 27 residual degree, 40, 140, 148 ring of integers, 40

semi-local, 210 semi-normal domain, 154 semi-normalization, 155 separation of points, 60, 62 Skolem

almost, 169 almost strong, 169, 296 closed, 161, 295 closure, 161, 295 ^-closure, 175 property, 160, 295

with respect to, 175 ring, 160

special chain, 101 Steinitz property, 212 Stone-Weierstrass

property, 55 theorem, 51, 57, 64

strong Skolem property, 160, 296 ring, 160

strongly n-generated, 208 strong 2-generator, 209 subset

arithmetically dense, 81 clopen, 52 fractional, 4, 286 polynomially

closed, 74 dense, 74 equivalent, 74

support, 18

torsion free-module, 15 module, 15 theory, 15

hereditary, 15 polynomial, 16

totally decomposed maximal ideal, 85 prime number, 82

two-generator property, 209

ultradistance, 52 ultrafilter, 119

322

ultrametric space, 66 unibranched domain, 63 unitary ideal, 124

fractional, 194

V.W.D.W.O. sequence, 29 valuation, 12

discrete, 13 domain, 13 essential, 8, 124 immediate, 145 normalized, 141 rank, 127 rank-one, 13

valuative dimension, 102 value

function, 198 ideal, 160, 198, 267, 287

Zariski ring, 76 topology, 208

zero divisor, 61

Selected Titles in This Series (Continued from the front of this publication)

12 J. R. Isbell, Uniform spaces, 1964 11 J. Cronin, Fixed points and topological degree in nonlinear analysis, 1964 10 R. Ayoub, An introduction to the analytic theory of numbers, 1963 9 Arthur Sard, Linear approximation, 1963 8 J. Lehner, Discontinuous groups and automorphic functions, 1964

7.2 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Volume II, 1961 7.1 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Volume I, 1961

6 C. C. Chevalley, Introduction to the theory of algebraic functions of one variable, 1951 5 S. Bergman, The kernel function and conformal mapping, 1950 4 O. F. G. Schilling, The theory of valuations, 1950 3 M. Marden, Geometry of polynomials, 1949 2 N. Jacobson, The theory of rings, 1943 1 J. A. Shohat and J. D. Tamarkin, The problem of moments, 1943