Graduate Texts in Mathematics 11 O Editorial Board

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

Springer Science+Business Media, LLC

BOOKS OF RELATED lNTEREST BY SERGE LANG

Linear Algebra, Third Edition 1987, ISBN 96412-6

Undergraduate Algebra, Second Edition 1990, ISBN 97279-X

Complex Analysis, Fourth Edition 1999, ISBN 98592-1

Real and Functional Analysis, Third Edition 1993, ISBN 94001-4

Introduction to Algebraic and Abelian Functions, Second Edition 1982, ISBN 90710-6

Cyclotomic Fields 1 and II 1990, ISBN 96671-4

0THER 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 • Introduction to Complex Hyper­bolic Spaces • Elliptic Functions • Number Theory III • Algebraic Number Theory • SL,(R) • Abelian V arieties • Differential and Riemannian Manifolds • Undergraduate Analysis • Elliptic Curves: Diophantine Analysis • lntroduction to Linear Algebra • Calculus of Severa! V ariables • 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

Serge Lang

Alge braic N um ber Theory

Second Edition

'Springer

Serge Lang Department of Mathematics Yale University New Haven, CT 06520 USA

Editorial Board

S. Axler Mathematics Department San Francisco State

University San Francisco, CA 94132 USA

F.W. Gehring Mathematics Department East Hali University of Michigan Ann Arbor, MI 48109 USA

K.A. Ribet Mathematics Department University of California

at Berkeley Berkeley, CA 94720-3840 USA

Mathematics Subject C!assifications (1991): llRxx, llSxx, llTxx

With 7 Illustrations

Library of Congress Cataloging-in-Publication Data Lang, Serge, 1927-

Algebraic number theory f Serge Lang. - 2nd ed. p. cm.- (Graduate texts in mathematics; 110)

Includes bibliographical references and index. ISBN 978-1-4612-6922-9 ISBN 978-1-4612-0853-2 (eBook) DOI 10.1007/978-1-4612-0853-2

1. Algebraic number theory. I. Title. II. Series. QA247.L29 1994 512'.74---dc20 93-50625

Originally published in 1970 © by Addison-Wesley Publishing Company, Inc., Reading, Massachusetts.

© 1994, 1986 by Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1986 Softcover reprint of the hardcover 2nd edition 1986

All rights reserved. This wark 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 forrner 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 7 6 5 4 3 (Corrected third printing )

ISBN 978-1-4612-6922-9 SPIN 10772455

Foreword

The present book gives an exposition of the classical basic algebraic and analytic number theory and supersedes my Algebraic Numbers, including much more material, e.g. the class field theory on which 1 make further comments at the appropriate place later.

For different points of view, the reader is encouraged to read the collec­tion of papers from the Brighton Symposium (edited by Cassels-Frohlich), the Artin-Tate notes on class field theory, Weil's book on Basic Number Theory, Borevich-Shafarevich's Number Theory, and also older books like those of W eber, Hasse, Hecke, and Hilbert's Zahlbericht. It seems that over the years, everything that has been done has proved useful, theo­retically or as examples, for the further development of the theory. Old, and seemingly isolated special cases have continuously acquired renewed significance, often after half a century or more.

The point of view taken here is principally global, and we deal with local fields only incidentally. For a more complete treatment of these, cf. Serre's book Corps Locaux. There is much to be said for a direct global approach to number fields. Stylistically, 1 have intermingled the ideal and idelic approaches without prejudice for either. 1 also include two proofs of the functional equation for the zeta function, to acquaint the reader with different techniques (in some sense equivalent, but in another sense, suggestive of very different moods). Even though a reader will prefer some techniques over alternative ones, it is important at least that he should be aware of all the possibilities.

New York June 1970

V

SERGE LANG

Preface for the Second Edition

The principal change in this new edition is a complete rewriting of Chapter XVII on the Explicit Formulas. Otherwise, I have made a few additions, and a number of corrections. The need for them was pointed out to me by severa! people, but I am especially indebted to Keith Conrad for the list he provided for me as a result of a very careful reading of the book.

New Haven, 1994 SERGE LANG

vi

Prerequisites

Chapters I through VII are self-contained, assuming only elementary algebra, say at the level of Galois theory.

Some of the chapters on analytic number theory assume some analysis. Chapter XIV assumes Fourier analysis on locally compact groups. Chap­ters XV through XVII assume only standard analytical facts (we even prove some of them), except for one allusion to the Plancherel formula in Chapter XVII.

In the course of the Brauer-Siegel theorem, we use the conductor­discriminant formula, for which we refer to Artin-Tate where a detailed proof is given. At that point, the use of this theorem is highly technical, and is due to the fact that one does not know that the zeros of the zeta function don't occur in a small interval to the left of 1. If one knew this, the proof would become only a page long, and the L-series 'vould not be needed at all. W e give Siegel's original proof for that in Chapter XIII.

My Algebra gives more than enough background for the present book. In fact, Algebra already contains a good part of the theory of integral extensions, and valuation theory, redone here in Chapters I and IL Furthermore, Algebra also contains whatever will be needed of group representation theory, used in a couple of isolated instances for applica­tions of the class field theory, or to the Brauer-Siegel theorem.

The word ring will always mean commutative ring without zero divisors and with unit element (unless otherwise specified).

If K is a field, then K* denotes its multiplicative group, and K its algebraic closure. Occasionally, a bar is also used to denote reduction modulo a prime ideal.

W e use the o and O notation. If /, g are two functions of a real variable, and g is always ~ O, we write f = O(g) if there exists a constant C > O such that lf(x)l ~ Cg(x) for all sufficiently large x. We writef = o(g) if lim"_..,f(x)/g(x) = O. We writef ~ g if lim"_..,f(x)jg(x) = 1.

vii

Contents

PartOne

General Basic Theory

CHAPTER I

Algebraic lntegers

1. Localization . 2. Integral closure 3. Prime ideals . 4. Chinese remainder theorem 5. Galois extensions 6. Dedekind rings 7. Discrete valuation rings 8. Explicit factorization of a prime 9. Projective modules over Dedekind rings

CHAPTER II

Completions

1. Definitions and completions 2. Polynomials in complete fields 3. Some filtrations . 4. U nramified extensions 5. Tamely ramified extensions

CHAPTER III

The Different and Discriminant

1. Complementary modules 2. The different and ramification 3. The discriminant

IX

3 4 8

11 12 18 22 27 29

31 41 45 48 51

57 62 64

X CONTENTS

CHAPTER IV

Cyclotomic Fields

1. Roots of unity 2. Quadratic fields 3. Gauss sums 4. Relations in ideal classes

CHAPTER V

Parallelotopes

1. The product formula 2. Lattice points in parallelotopes 3. A volume computation 4. Minkowski's constant

CHAPTER VI

The Ideal Function

1. Generalized ideal classes 2. Lattice points in homogeneously expanding domains 3. The number of ideals in a given class .

CHAPTER VII

ldeles and Adeles

1. Restricted direct products 2. Adeles . 3. Ideles . 4. Generalized ideal class groups; relations with idele classes 5. Embedding of k: in the idele classes . 6. Galois operation on ideles and idele classes

CHAPTER VIII

Elementary Properties of the Zeta Function and L-series

1. Lemmas on Dirichlet series . 2. Zeta function of a number field 3. The L-series . 4. Density of primes in arithmetic progressions . 5. Faltings' finiteness theorem

71 76 82 96

99 110 116 119

123 128 129

137 139 140 145 151 152

155 159 162 166 170

CONTENTS

PartTwo

Class Field Theory

CHAPTER IX

Norm Index Computations

1. Algebraic preliminaries . 2. Exponential and logarithm functions 3. The local norm index 4. A theorem on units . 5. The global cyclic norm index 6. Applications .

CHAPTER X

The Artin Symbo1, Reciprocity Law, and C1ass Fie]d Theory

1. Formalism of the Artin symbol 2. Existence of a conductor for the Artin symbol 3. Class fields

CHAPTER XI

The Existence Theorem and Local CJass Field Theory

1. Reduction to Kummer extensions 2. Proof of the existence theorem . 3. The complete splitting theorem 4. Local class field theory and the ramification theorem 5. The Hilbert class field and the principal ideal theorem 6. Infinite divisibility of the universal norms

CHAPTER XII

L-series Again

1. The proper abelian L-series 2. Artin (non-abelian) L-series 3. Induced characters and L-series contributions

xi

179 185 187 190 193 195

197 200 206

213 215 217 219 224 226

229 232 236

xii CONTENTS

Part Three

Analytic Theory

CHAPTER XIII

Functional Equation of the Zeta Function, Hecke's Proof

1. The Poisson summation formula 245 2. A special computation 250 3. Functional equation . 253 4. Application to the Brauer-Siegel theorem 260 5. Applications to the ideal function 262

Appendix: Other applications 273

CHAPTER XIV

Functional Equation, Tate's Thesis

1. Local additive duality 276 2. Local multiplicative theory . 278 3. Local functional equation 280 4. Local computations 282 5. Restricted direct products 287 6. Global additive duality and Riemann-Roch theorem 289 7. Global functional equation 292 8. Global computations 297

CHAPTER XV

Density of Primes and Tauberian Theorem

1. The Dirichlet integral 303 2. Ikehara's Tauberian theorem 304 3. Tauberian theorem for Dirichlet series 310 4. N on-vanishing of the L-series 312 5. Densities 315

CHAPTER XVI

The Brauer-Siegel Theorem

1. An upper estimate for the residue . 322 2. A lower bound for the residue 323 3. Comparison of residues in normal extensions 325 4. End of the proofs 327

Appendix: Brauer's lemma 328

CONTEN'fS

CHAPTER XVII

Explicit Formulas

1. Weierstrass factorization of the L-series 2. An estimate for ~' / ~ . 3. The Weil formula 4. The basic sum and the first part of its evaluation 5. Evaluation of the sum: Second part

Bibliography

Index .

XIII

331 333 337 344 348

353

355