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View This Volume's Front and Back Matter http://dx.doi.org/10.1090/gsm/024 Selected Titles in This Series 24 Helmut Koch, Number Theory: Algebraic Numbers and Functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997 15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997 14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing 10 Barry Simon, Representations of finite and compact groups, 1996 9 Dino Lorenzini, An invitation to arithmetic geometry, 1996 8 Winfried Just and Martin Weese, Discovering modern set theory. I: The basics, 1996 7 Gerald J. Janusz, Algebraic number fields, second edition, 1996 6 Jens Carsten Jantzen, Lectures on quantum groups, 1996 5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995 4 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994 3 William W. Adams and Philippe Loustaunau, An introduction to Grobner bases, 1994 2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity, 1993 1 Ethan Akin, The general topology of dynamical systems, 1993 This page intentionally left blank Number Theory Algebraic Numbers and Functions This page intentionally left blank Numbe r Theor y Algebraic Numbers and Functions Helmut Koch Translated by David Kramer Graduate Studies in Mathematics Volume 24 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE James Humphreys (Chair) David Saltman David Sattinger Ronald Stern Originally published in the German language by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, D-65189 Wiesbaden, Germany, under the title "Helmut Koch: Zahlentheorie. Algebraische Zahlen und Funktionen. 1. Auflage (1st edition)". © by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1997 Translated from the German by David Kramer 2000 Mathematics Subject Classification. Primary llRxx, llSxx, llMxx. ABSTRACT. The primary goal of this book is to present the essential elements of algebraic number theory, including the theory of normal extensions up through a glimpse of class field theory. Following the example set by Kronecker, Weber, Hilbert and Artin, algebraic functions are handled on an equal footing as algebraic numbers. This is done on the one hand to demonstrate the analogy between number fields and function fields, which is especially strong in the case where the ground field is a finite field. On the other hand, in this way one obtains an introduction to the theory of 'higher congruences' as an important element of 'arithmetic geometry'. The book is suitable for two one-semester courses, leading from basic algebraic number theory up to the beginning of class field theory. Library of Congress Cataloging-in-Publication Data Koch, Helmut, 1932- [Zahlentheorie. English] Number theory : algebraic numbers and functions / Helmut Koch. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 24) Includes bibliographical references and index. ISBN 0-8218-2054-0 (acid-free paper) 1. Number theory. I. Title. II. Series. QA241 .K67713 2000 512/.7-dc21 00-022320 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 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. © 2000 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. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 05 04 03 02 01 00 Contents Preface xiii Translator's Note xvi Notation xvii List of Symbols xvii Chapter 1. Introduction 1 1.1. Pythagorean Triples 1 1.2. Pell's Equation 3 1.3. Fermat's Last Theorem 4 1.4. Congruences 8 1.5. Public Key Cryptology 11 1.6. Quadratic Residues 12 1.7. Prime Numbers 22 1.8. The Prime Number Theorem 26 1.9. Exercises 31 Chapter 2. The Geometry of Numbers 35 2.1. Binary Quadratic Forms 35 2.2. Complete Decomposable Forms of Degree n 37 2.3. Modules and Orders 39 vii Vlll Contents 2.4. Complete Modules in Finite Extensions of P 43 2.5. The Integers of a Quadratic Field 45 2.6. Further Examples of Determining a Z-Basis for the Ring of Integers of a Number Field 46 2.7. The Finiteness of the Class Number 47 2.8. The Group of Units 48 2.9. The Start of the Proof of Dirichlet's Unit Theorem 50 2.10. The Rank of 1 (E) 51 2.11. The Regulator of an Order 55 2.12. The Lattice Point Theorem 55 2.13. Minkowski's Geometry of Numbers 57 2.14. Application to Complete Decomposable Forms 62 2.15. Exercises 64 Chapter 3. Dedekind's Theory of Ideals 65 3.1. Basic Definitions 66 3.2. The Main Theorem of Dedekind's Theory of Ideals 68 3.3. Consequences of the Main Theorem 71 3.4. The Converse of the Main Theorem 73 3.5. The Norm of an Ideal 74 3.6. Congruences 76 3.7. Localization 78 3.8. The Decomposition of a Prime Ideal in a Finite Separable Extension 80 3.9. The Class Group of an Algebraic Number Field 84 3.10. Relative Extensions 88 3.11. Geometric Interpretation 93 3.12. Different and Discriminant 94 3.13. Exercises 101 Contents IX Chapter 4. Valuations 103 4.1. Fields with Valuation 104 4.2. Valuations of the Field of Rational Numbers and of a Field of Rational Functions 110 4.3. Completion 112 4.4. Complete Fields with Respect to a Discrete Valuation 114 4.5. Extension of a Valuation of a Complete Field to a Finite Extension 121 4.6. Finite Extensions of a Complete Field with a Discrete Valuation 124 4.7. Complete Fields with a Discrete Valuation and Finite Residue Class Field 129 4.8. Extension of the Valuation of an Arbitrary Field to a Finite Extension 132 4.9. Arithmetic in the Compositum of Two Field Extensions 137 4.10. Exercises 137 Chapter 5. Algebraic Functions of One Variable 141 5.1. Algebraic Function Fields 142 5.2. The Places of an Algebraic Function Field 144 5.3. The Function Space Associated to a Divisor 149 5.4. Differentials 154 5.5. Extensions of the Field of Constants 158 5.6. The Riemann-Roch Theorem 160 5.7. Function Fields of Genus 0 164 5.8. Function Fields of Genus 1 167 5.9. Exercises 169 Chapter 6. Normal Extensions 171 6.1. Decomposition Group and Ramification Groups 172 6.2. A New Proof of Dedekind's Theorem on the Different 176 6.3. Decomposition of Prime Ideals in an Intermediate Field 178 6.4. Cyclotomic Fields 180 X Contents 6.5. The First Case of Fermat's Last Theorem 184 6.6. Localization 188 6.7. Upper Numeration of the Ramification Group 190 6.8. Kummer Extensions 195 6.9. Exercises 199 Chapter 7. L-Series 203 7.1. From the Riemann ("-Function to the Hecke L-Series 204 7.2. Normalized Valuations 207 7.3. Adeles 209 7.4. Ideles 212 7.5. Idele Class Group and Ray Class Group 214 7.6. Hecke Characters 217 7.7. Analysis on Local Additive Groups 219 7.8. Analysis on the Adele Group 223 7.9. The Multiplicative Group of a Local Field 227 7.10. The Local Functional Equation 230 7.11. Calculation of p(c) for K = R 232 7.12. Calculation of p{c) for K = C 234 7.13. Computation of the p-Factors for a Nonarchimedean Field 236 7.14. Relations Among the p-Factors 239 7.15. Analysis on the Idele Group 240 7.16. Global Zeta Functions 243 7.17. The Dedekind Zeta Function 247 7.18. Hecke L-Series 251 7.19. Congruence Zeta Functions 252 7.20. Exercises 257 Chapter 8. Applications of Hecke L-Series 259 8.1. The Decomposition of Prime Numbers in Algebraic Number Fields 259 8.2. The Nonvanishing of the L-Series at s = 1 262 Contents XI 8.3. The Distribution of Prime Ideals in an Algebraic Number Field 266 8.4. The Generalized Riemann Hypothesis 270 8.5. Exercises 273 Chapter 9. Quadratic Number Fields 275 9.1. Quadratic Forms and Orders in Quadratic Number Fields 275 9.2. The Class Number of Imaginary Quadratic Number Fields 282 9.3.
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