Graduate Texts in Mathematics 84Editorial Board

s. Axler F.W. Gehring K.A. Ribet

Springer Science+Business Media, LLC

Graduate Texts in Mathematics

TAKEUTtlZARING. Introduction to 34 SPITZER. Principles of Random Walk.Axiomatic Set Theory.2nd ed. 2nd ed.

2 OXTOBY. Measure and Category. 2nd ed. 35 ALEXANDERIWERMER. Several Complex3 SCHAEFER. Topological Vector Spaces. Variables and Banach Algebras. 3rd ed.

2nd ed. 36 KELLEy!NAMIOKA et al. Linear4 HILTON/STAMMBACH. A Course in Topological Spaces.

Homological Algebra. 2nd ed. 37 MONK. Mathematical Logic.5 MAC LANE. Categories for the Working 38 GRAUERT/FRlTZSCHE. Several Complex

Mathematician. 2nd ed. Variables.6 HUGHES/PIPER. Projective Planes. 39 ARVESON. An Invitation to C*-Algebras.7 J.-P. SERRE. A Course in Arithmetic. 40 KEMENy/SNELLlKNAPP. Denumerable8 TAKEUTtlZARlNG. Axiomatic Set Theory. Markov Chains. 2nd ed.9 HUMPHREYS. Introduction to Lie Algebras 41 ApOSTOL. Modular Functions and

and Representation Theory. DirichletSeries in NumberTheory.10 COHEN. A Course in Simple Homotopy 2nd ed.

Theory. 42 J.-P. SERRE. Linear Representations ofII CONWAY. Functions of One Complex Finite Groups.

Variable I. 2nd ed. 43 GILLMAN/JERISON. Rings of Continuous12 BEALS. AdvancedMathematical Analysis. Functions.13 ANDERSON/FuLLER. Rings and Categories 44 KENDIG. ElementaryAlgebraic Geometry.

of Modules. 2nd ed. 45 LoEVE. Probability Theory I. 4th ed,14 GOLUBITSKy/GUILLEMIN. Stable Mappings 46 LOEVE. Probability Theory [I. 4th ed.

and Their Singularities. 47 MOISE. GeometricTopologyin15 BERBERIAN. Lectures in Functional Dimensions 2 and 3.

Analysis and OperatorTheory. 48 SACHslWu. General Relativity for[6 WINTER. The Structure of Fields. Mathematicians.17 ROSENBLATT. Random Processes. 2nd ed. 49 GRUENBERoIWEIR. Linear Geometry.18 HALMOS. Measure Theory. 2nd ed.19 HALMOS. A HilbertSpace Problem Book. 50 EDWARDS. Fermat's LastTheorem.

2nd ed. 51 KuNGENBERG. A Course in Differential20 HUSEMOLLER. Fibre Bundles. 3rd ed. Geometry.21 HUMPHREYS. Linear Algebraic Groups. 52 HARTSHORNE. Algebraic Geometry.22 BARNEs/MACK. An Algebraic Introduction 53 MANIN. A Course in Mathematical Logic.

to Mathematical Logic. 54 GRAVER/WATKINS. Combinatorics with23 GREUB. LinearAlgebra. 4th ed. Emphasis on the Theoryof Graphs.24 HOLMES. Geometric Functional Analysis 55 BROWN/PEARCY. Introduction to Operator

and Its Applications. Theory I: Elementsof Functional Analysis.25 HEWITT/STROMBERG. Realand Abstract 56 MASSEY. Algebraic Topology: An

Analysis. Introduction.26 MANES. Algebraic Theories. 57 CROWELUFox. Introduction to Knot27 KELLEY. GeneralTopology. Theory.28 ZARISKJ/SAMUEL. Commutative Algebra. 58 KOBLITZ. p-adic Numbers.p-adic

Vol.I. Analysis, and Zeta-Functions. 2nd ed.29 ZARISKJ/SAMUEC Commutative Algebra. 59 LANG. Cyclotomic Fields.

Vol.II. 60 ARNOLD. Mathematical Methods in30 JACOBSON. Lecturesin AbstractAlgebra I. Classical Mechanics. 2nd ed.

BasicConcepts. 61 WHITEHEAD. Elementsof Homotopy31 JACOBSON. l,ectures in AbstractAlgebra Il. Theory.

LinearAlgebra. 62 KARGAPOLOvIMERLZJAKOV. Fundamentals32 JACOBSON Lectures in Abstract Algebra of the Theoryof Groups.

U1. Theoryof Fields and GaloisTheory. 63 BOLLOBAS. Graph Theory:33 HIRSCH. DifferentialTopology.

(continued after index)

Kenneth Ireland

Michael Rosen

A Classical Introduction toModern Number Theory

Second Edition

Springer

Kenneth Ireland(deceased)

Editorial Board

S. AxlerMathematics DepartmentSan Francisco State

UniversitySan Francisco, CA 94132USA

With I illustration.

F.W. GehringMathematics DepartmentEast HallUniversity of MichiganAnn Arbor, MI 48109USA

Michael RosenDepartment of MathematicsBrown UniversityProvidence , RI 02912USA

K.A . RibetDepartment of MathematicsUniversity of California

at BerkeleyBerkeley, CA 94720·3840USA

Mathematics SubjectClassification (2000): II..QI, 11-02

Library of Congress Cataloging-in-Publication DataIreland, Kenneth F.

A classical introduction to modem number theory / KennethIreland, Michael Rosen.-2nd ed .

p. cm.-(Graduate texts in mathematics; 84)Include s bibliographical references and index .I. Number theory . I. Rosen , Michael I. II. Title. III . Series .

QA241.I667 1990512.7-dc20

90-9848Printed on acid-free paper .

"A Classical Introduction to Modem Number Theory" is a revised and expanded version of.. Elements of Number Theory " published in 1972by Bogden and Quigley, Inc. , PUbl ishers.

©1972, 1982, 1990 Springer Science+Business Media New YorkOriginallypublished by Springer-Verlag New York, Inc. in 1990.

All rights reserved. This work may not be translated or copied in whole or in part without thewritten permission of the publisher Springer Science+Business Media, LLC, except for briefexcerpts in connection with reviews or scholarly analysis. Use in connection with anyform of information storage and retrieval, electronic adaptation , computer software, orby 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 Merchand ise Marks Act, may accordingly be usedfreely by anyone .

This reprinthas beenauthorizedby Springer-Verlag (Beriin/HeidelberglNew York) forsale inthe People's Republicof China onlyand not forexport therefrom.Reprinted in China by BeijingWorldPublishingCorporation, 2003

987

ISBN 978-1-4419-3094-1 ISBN 978-1-4757-2103-4 (eBook)DOI 10.1007/978-1-4757-2103-4

Preface to the Second Edition

It is now 10 years since the first edit ion of this book appeared in 1980. Theintervening decade has seen tremendous advances take place in mathe­matic s generally, and in number theory in particular. It would seem desir­able to treat some of these advances , and with the addition of two newchapter s, we are able to cover some portion of this new material.

As examples of important new work that we have not included, wemention the following two results:

(I) The first case of Fermat's last theorem is true for infinitely manyprime exponents p. This means that, for infinitely many primes p , theequ ation x P + yP = zP has no solutions in nonzero integers with p r.ryz. Th is was proved by L.M. Adelman and D.R. Heath-Brown andindependently by E. Fouvry . An overview of the proof is given byHeath-Brown in the Mathematical Intellig encer (Vol. 7, No.6, 1985).

(2) Let PI , P2, and P3 be three distinct primes . Then at least one of them isa primitive root for infinitely many primes q. Recall that E. Artinconjectured that, if a E 7L is not 0, I, - I, or a square, then there areinfinitely many primes q such that a is a primitive root modulo q. Thetheorem we have stated was proved in a weaker form by R. Gupta andM.R. Murty, and then strengthened by the combined efforts of R.Gupta, M.R . Murty, V.K . Murty , and D.R. Heath-Brown. An exposi­tion of this result, as well as an analogue on elliptic curves, is given byM.R . Murty in the Mathematic:allntelligencer (Vol. 10, No.4, 1988).

The new material that we have added falls principally within the frame­work of arithmetic geometry. In Chapter 19 we give a complete proof ofL.J. Mordell's fundamental theorem , which asserts that the group of ra­tional points on an elliptic curve, defined over the rational numbers, isfinitely generated . In keeping with the spirit of the book, the proof (due inessence to A. Weil) is elementary. It makes no use of cohomology groupsor any other advanced machinery . It does use finiteness of class numberand a weak form of the Dirichlet unit theorem; both results are proved inthe text.

The second new chapter, Chapter 20, is an overview of G. Faltings 'sproof of the Mordell conjecture and recent progress on the arithmetic of

v

vi Preface to the Second Edition

elliptic curves, especially the work of B. Gross , V.A. Kolyvagin , K.Rubin , and D. Zagier. Some of this work has surprising applications toother areas of number theory. We discuss one application to Fermat's lasttheorem , due to G. Frey, J.P. Serre, and K. Ribet. Another importantapplication is the solution of an old problem due to K .F . Gauss aboutclass numbers of imaginary quadratic number fields. This comes about bycombining the work of B. Gross and D. Zagier with a result of D. Gold­feld. This chapter contains few proofs. Its main purpose is to give aninformative survey in the hope that the reader will be inspired to learn thebackground necessary to a better understanding and appreciation of theseimportant new developments .

The rest of the book is essentially unchanged . An attempt has beenmade to correct errors and misprints. In an effort to keep confusion to aminimum, we have not changed the bibliography at the end of the book .New references for the two new chapters, Chapters 19 and 20, will befound at the end of those chapters . We would like to thank Tom Nakaharaand others for submitting a list of misprints from the first edition. Also, wethank Linda Guthrie for typing portions of the final chapters.

We have both been very pleased with the warm reception that the firstedition of this book received . It is our hope that the new edition willcontinue to entice readers to delve deeper into the mysteries of this an­cient, beautiful, and still vital subject.

February 1990 Kenneth IrelandMichael Rosen

Addendum to Second Edition . Second Corrected Printing

The second printing of the second edit ion is unchanged except for correc­tions and the addition of a few clarifying comments. I would like to thankK. Conrad, M. Jastrzebski, F. Lemmermeyer and others who took thetrouble to send us detailed lists of misprints.

No vember 1992 Michael Rosen

Notes for the Second Edition, Fifth Corrected Printing

In 1995 Andrew Wiles published a paper in the Annals of Mathematicswhich proved the Taniyarna-Shimura-Weil conjecture is true for semi-stableelliptic curves over the rational numbers. Together with earlier results, prin­cipally the theorem of Ken Ribet mentioned on page 347, this provedFermat's Last Theorem. The most famous conjecture in elementary numbertheory is finally a theorem!!!

April 1998 Michael Rosen

Preface

This book is a revised and greatly expanded version of our book Elements ofNumber Theory published in 1972. As with the first book the primary audiencewe envisage consists of upper level undergraduate mathematics majors andgraduate students. We have assumed some familiarity with the material in astandard undergraduate course in abstract algebra. A large portion ofChapters 1-11 can be read even without such background with the aid of asmall amount of supplementary reading. The later chapters assume someknowledge of Galois theory, and in Chapters 16 and 18 an acquaintance withthe theory of complex variables is necessary.

Number theory is an ancient subject and its content is vast. Any intro­ductory book must, of necessity, make a very limited selection from thefascinating array of possible topics. Our focus is on topics which point in thedirection of algebraic number theory and arithmetic algebraic geometry. Byacareful selection of subject matter we have found it possible to exposit somerather advanced material without requiring very much in the way of technicalbackground. Most of this material is classical in the sense that is was dis­covered during the nineteenth century and earlier, but it is also modernbecause it is intimately related to important research going on at the presenttime.

In Chapters 1-5 we discuss prime numbers, unique factorization, arith­metic functions, congruences, and the law of quadratic reciprocity. Very littleis demanded in the way of background. Nevertheless it is remarkable how amodicum of group and ring theory introduces unexpected order into thesubject. For example, many scattered results turn out to be parts of the answerto a natural question: What is the structure of the group of units in the ring7L/n7L ?

Reciprocity laws constitute a major theme in the later chapters. The lawof quadratic reciprocity, beautiful in itself, is the first of a series of reciprocitylaws which lead ultimately to the Artin reciprocity law. one of the majorachievements of algebraic number theory. We travel along the road beyondquadratic reciprocity by formulating and proving the laws of cubic andbiquadratic reciprocity. In preparation for this many of the techniques ofalgebraic number theory are introduced; algebraic numbers and algebraicintegers, finite fields, splitting of primes, etc. Another important tool in this

vii

viii Preface

investigation (and in others!) is the theory of Gauss and Jacobi sums. Thismaterial is covered in Chapters 6-9. Later in the book we formulate and provethe more advanced partial generalization of these results, the Eisensteinreciprocity law.

A second major theme is that of diophantine equations, at first over finitefields and later over the rational numbers. The discussion of polynomialequations over finite fields is begun in Chapters 8 and 10 and culminates inChapter II with an exposition of a portion of the paper" Number of solutionsof equations over finite fields " by A. Weil. This paper, published in 1948, hasbeen very influential in the recent development of both algebraic geometryand number theory. In Chapters 17and 18we consider diophantine equationsover the rational numbers. Chapter 17 covers many standard topics fromsums of squares to Fermat's Last Theorem. However, because of materialdeveloped earlier we are able to treat a number of these topics from a novelpoint of view. Chapter 18 is about the arithmetic of elliptic curves. It dif­fers from the earlier chapters in that it is primarily an overview with manydefinitions and statements of results but few proofs. Nevertheless, by con­centrating on some important special cases we hope to convey to the readersomething of the beauty of the accomplishments in this area where much workis being done and many mysteries remain.

The third, and final, major theme is that of zeta functions. In Chapter 11 wediscuss the congruence zeta function associated to varieties defined over finitefields. In Chapter 16 we discuss the Riemann zeta function and the DirichletL-functions. In Chapter 18 we discuss the zeta function associated to analgebraic curve defined over the rational numbers and Heeke L-functions.Zeta funct ions compress a large amount of arithmetic information into asingle function and make possible the application of the powerful methods ofanalysis to number theory.

Throughout the book we place considerable emphasis on the history of.our subject. In the notes at the end of each chapter we give a brief historicalsketch and provide references to the literature. The bibliography is extensivecontaining many items both classical and modern. Our aim has been toprovide the reader with a wealth of material for further study.

There are many exercises, some routine, some challenging. Some of theexercises supplement the text by providing a step by step guide through theproofs of important results. In the later chapters a number of exercises havebeen adapted from results which have appeared in the recent literature. Wehope that working through the exercises will be a source of enjoyment as wellas instruction.

In the writing of this book we have been helped immensely by the interestand assistance of many mathematical friends and acquaintances. We thankthem all. In particular we would like to thank' Henry Pohlmann who insistedwe follow certain themes to their logical conclusion, David Goss for allowingus to incorporate some of his work into Chapter 16, and Oisin McGuinessfor his invaluable assistance in the preparation of Chapter 18. We would

Preface ix

like to thank Dale Cavanaugh, Janice Phillips, and especially Carol Ferreira,for their patience and expertise in typing large portions of the manuscript.Finally, the second author wishes to express his gratitude to the VaughnFoundation Fund for financial support during his sabbatical year inBerkeley , California (1979/80).

July 25, 1981 Kenneth IrelandMichael Rosen

Contents

Preface to the Second EditionPreface

CHAPTER 1Unique Factorization

§I Unique Factorization in 7l.§2 Unique Factorization in k[x]§3 Unique Factorization in a Principal Ideal Domain§4 The Rings 7l. [i ) and 7l. [w) .

CHAPTER 2Applications of Unique Factorization

§ I Infinitely Many Primes in 7l.§2 Some Arithmetic Functions§3 L lip Diverges§4 The Growth of 1T(X)

CHAPTER 3Congruence

§ I Elementary Observations§2 Congruence in 7l.§3 The Congruence ax == b(m)§4 The Chinese Remainder Theorem

CHAPTER 4The Structure of U("l./n"l.)

§ I Primitive Roots and the Group Structure of U(71.1 n71.)§2 nth Power Residues

CHAPTER 5Quadratic Reciprocity

§ I Quadratic Residues§2 Law of Quadratic Reciprocity§3 A Proof of the Law of Quadratic Reciprocity

vvii

1

I68

12

17

17182122

2828293134

39

3945

50

505358

xi

xii

CHAPTER 6Quadratic Gauss Sums

§ I Algebraic Numbers and Algebraic Integers§2 The Quadratic Character of 2§3 Quadratic Gauss Sums§4 The Sign of the Quadratic Gauss Sum

CHAPTER 7Finite Fields

§ I Basic Properties of Finite Fields§2 The Existence of Finite Fields§3 An Application to Quadratic Residues

CHAPTER 8Gauss and Jacobi Sums

§ I Multipl icative Characters§2 Gauss Sums§3 Jacobi Sums§4 The Equation x" + y" = I in Fp

§5 More on Jacobi Sums§6 Applications§7 A General Theorem

CHAPTER 9Cubic and Biquadratic Reciprocity

§I The Ring l[w]§2 Residue Class Rings§3 Cubic Residue Character§4 Proof of the Law of Cubic Reciprocity§5 Another Proof of the Law of Cubic Reciprocity§6 The Cubic Character of 2§7 Biquadratic Reciprocity : Preliminaries§8 The Quartic Residue Symbol§9 The Law of Biquadratic Reciprocity

§10 Rational Biquadratic Reciprocity§II The Constructibility of Regular Polygons§ 12 Cub ic Gauss Sums and the Problem of Kummer

CHAPTER toEquations over Finite Fields

§I Affine Space, Projective Space, and Polynomials§2 Chevalley 's Theorem§3 Gauss and Jacobi Sums over Finite Fields

Contents

6666697073

79

798385

88

8891.929798

101102

108

109III112115117118119121123127130131

138

138143145

Contents xiii

CHAPTER I [The Zeta Function 151

§I The Zeta Function of a Projective Hypersurface 151§2 Trace and Norm in Finite Fields 158§3 The Rationality of the Zeta Function Associated to

aoxo + a\xj + .. . + anx': 161§4 A Proof of the Hasse-Davenport Relation 163§5 The Last Entry 166

CHAPTER 12Algebraic Number Theory 172

§ I Algebraic Preliminaries 172§2 Unique Factorization in Algebraic Number Fields 174§3 Ramification and Degree 181

CHAPTER 13Quadratic and Cyclotomic Fields 188

§I Quadratic Number Fields 188§2 Cyclotomic Fields 193§3 Quadratic Reciprocity Revisited 199

CHAPTER 14The Stickelberger Relation and the Eisenstein Reciprocity Law 203

§ I The Norm of an Ideal 203§2 The Power Residue Symbol 204§3 The Stickelberger Relation 207§4 The Proof of the Stickelberger Relation 209§5 The Proof of the Eisenstein Reciprocity Law 215§6 Three Applications 220

CHAPTER 15Bernoulli Numbers 228

§I Bernoulli Numbers ; Definitions and Applications 228§2 Congruences Involving Bernoulli Numbers 234§3 Herbrand's Theorem 241

CHAPTER 16Dirichlet L-functions 249

§ I The Zeta Function 249§2 A Special Case 251§3 Dirichlet Characters 253§4 Dirichlet L-functions 255§5 The Key Step 257§6 Evaluating Lis, X) at Negative Integers 261

xiv

CHAPTER 17Diophantine Equations

§ I Generalities and First Examples§2 The Method of Descent§3 Legendre's Theorem§4 Sophie Germain 's Theorem§5 Pelt's Equation§6 Sums of Two Squares§7 Sums of Four Squares§8 The Fermat Equation: Exponent 3§9 Cubic Curves with Infinitely Many Rational Points

§1O The Equation y 2 = xl + k§II The First Case of Fermat's Conjecture for Regular Exponent§12 Diophantine Equations and Diophantine Approximation

CHAPTER 18Elliptic Curves

§ I Generalities§2 Local and Global Zeta Functions of an Elliptic Curve§3 y2 = xl + D, the Local Case§4 y 2 = Xl - Dx, the Local Case§5 Heeke L-functions§6 y2 = x l - Dx, the Global Case§7 y 2 = Xl + D, the Global Case§8 Final Remarks

CHAPTER 19The Mordell-Weil Theorem

§ I The Addition Law and Several Identities§2 The Group EI2E§3 The Weak Dirichlet Unit Theorem§4 The Weak Mordell-Weil Theorem§5 The Descent Argument

CHAPTER 20New Progress in Arithmetic Geometry

§I The Mordell Conjecture§2 Elliptic Curves§3 Modular Curves§4 Heights and the Height Regulator§5 New Results on the Birch-Swinnerton-Dyer Conjecture§6 Applications to Gauss's Class Number Conjecture

Selected Hints for the Exercises

Bibliography

Index

Contents

269269271272275276278280284287288290292

297297301304306307310312314

319

320323326328330

339

340343345348353358

367

375

385