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BOOKS OF RELATED INTEREST BY SERGE LANG Linear Algebra, Third Edition 1987, ISBN 96412-6 Undergraduate Algebra, Second Edition 1990, ISBN 97279-X Complex Analysis, Third Edition 1993, ISBN 97886-0 Real and Functional Analysis, Third Edition 1993, ISBN 94001-4 Introduction to Algebraic and Abelian Functions, Second Edition 1982, ISBN 90710-6 Cyclotomic Fields I and II 1990, ISBN 96671-4 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' Introdm:tion to Complex Hyper bolic Spaces • Elliptic Functions • Number Theory III • Algebraic Number Theory • SL2(R) • Abelian Varieties • Differential and Riemannian Manifolds • Undergraduate Analysis • 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

Serge Lang I /Algebraic Number Theory/ / Second Edition , I ' 1 ; :'" . . \ . , !" >- Springer

Serge Lang Department of Mathematics Yale University N"ew Haven, CT 06520 USA Editorial Board S. Axler D~partment of Mathematics F.W. Gehring Department of Mathematics P.R. Halmos Department of Mathematics Michigan State University Eest Lansing, MI 48824 USA University of Michigan Ann Arbor, MI 48109 USA Santa Clara University Santa Clara, CA 95053 USA Mathematics Subject Classifications (1991): 11Rxx, llSxx, llTxx With 7 Illustrations Library of Congress Cataloging-in-Publication Data Lang, Serge, 1927- j Algebraic number theory! Serge Lang. - 2nd ed. (Graduate texts in mathematics; 110) em. - p. Includes bibliographical references and index. ISBN 0-387-94225-4 1. Algebraic number theory. 1. Title. '. QA247.L29 1994 51Z'.74-dc20 II. Series. 93-506Z!fo' Originally published in 1970 © by Addison-Wesley Publishing Company, Inc., Reading, Massachusetts. © 1894, 1986 by Springer-Verlag New York, Inc. All rights reserved. This werk may not be translated or copied in whole or in part without the ~Titten permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), 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. Printed and bound by Edwards Brothers, Ann Arbor, MI. Printed in the United States of America. 98765 432 ISBN 0-387-94225-4 Springer-Verlag New York Berlin Heidelberg ISBN 3-540-94225-4 Springer-Verlag Berlin Heidelberg New York

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 I make further comments at the appropriate place later. For different points of view, the reader is encouraged to read the collec tion 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 It seems that those of Weber, Hasse, Heeke, and Hilbert's Zahlbericht. over the years, everything that has been done has proved useful, theo retically 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, I have intermingled the ideal and idelic approaches without prejudice for either. I 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 SERGE LAKG v

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 several 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. Chap ters 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 conductor discriminant 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 would not be needed at all. We 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 II. Furthermore, Algebra also contains whatever will be needed of group representation theory, used in a couple of isolated instances for applica tions 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 Kits algebraic closure. Occasionally, a bar is also used to denote reduction modulo a prime ideal. We use the 0 and 0 notation. If j, g are two functions of a real variable, and g is always ~ 0, we write j = O(g) if there exists a constant C > 0 such that Ij(x)I ~ Cg(x) for all sufficiently large x. We write j = o(g) if lim",--->",j(x)jg(x) = O. We writej ~ g iflim",--->",j(x)jg(x) = 1. vii

Contents Part One General Basic Theory CHAPTER I Algebraic Integers -\.1. 2. 3. 4. 5. 6. 7. 8. 9. . Localization Integral closure Prime ideals . Chinese remainder theorem Galois extensions Dedekind rings Discrete valuation rings Explicit factorization of a prime Projective modules over Dedekind rings CHAPTER II Completions -!.. 1. Definitions and completions Polynomials in complete fields 2. 3. Some filtrations 4. Unramified 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 l. Roots of unity 2. Quadratic fields 3. Gauss sums 4. Relations in ideal classes Cor,TENTS CHAPTER IV Cyclotomic Fields CHAPTER V Parallelotopes l. The product formula 2. Lattice points in parallelotopes 3. A volume computation 4. Minkowski's constant CHAPTER VI The Ideal Function l. Generalized ideal classes 2. Lattice points in homogeneously expanding domains 3. The number of ideals in a given class . CHAPTER VII Ideles and Adeles '- l. 2. ~ 3. 4. 5. 6. Restricted direct products Adeles . Ideles . Generalized ideal class groups; relations with idele classes Embedding of k': in the idele classes Galois operation on ideles and idele classes . CHAPTER VIII Elementary Properties of the Zeta Function and L-series l. 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

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