Graduate Texts in Mathematics 121 Editorial Board J.H Ewing F.W Gehring P.R Halmos Graduate Texts in Mathematics 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 TAKEUTIIZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra MAC LANE Categories for the Working Mathematician HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic T AKEUTIIZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FuLLER Rings and Categories of Modules GOLUBITSKy/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMos Measure Theory HALMos A Hilbert Space Problem Book 2nd ed., revised HUSEMOLLER Fibre Bundles 2nd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKIISAMUEL Commutative Algebra Vol I ZARISKIISAMUEL Commutative Algebra Vol II JACOBSON Lectures in Abstract Algebra I: Basic Concepts JACOBSON Lectures in Abstract Algebra II: Linear Algebra JACOBSON Lectures in Abstract Algebra III: Theory of Fields and Galois Theory HIRSCH Differential Topology SPITZER Principles of Random Walk 2nd ed WERMER Banach Algebras and Several Complex Variables 2nd ed KELLEy/NAMIOKA et al Linear Topological Spaces MONK Mathematical Logic GRAUERT/FRITZSCHE Several Complex Variables ARVESON An Invitation to C*-Algebras KEMENy/SNELL/KNAPP Denumerable Markov Chains 2nd ed APOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed SERRE Linear Representations of Finite Groups GILLMAN/JERISON Rings of Continuous Functions KENDIG Elementary Algebraic Geometry LOEVE Probability Theory I 4th ed LOEVE Probability Theory II 4th ed MOISE Geometric Topology in Dimensions and Serge Lang Cyclotomic Fields I and II Combined Second Edition With an Appendix by Karl Rubin Springer Science+Business Media, LLC Serge Lang Department of Mathematics Yale University New Haven, CT 06520 U.S.A Editorial Board J.H Ewing Department of Mathematics Indiana University B1oomington, IN 47405 U.SA F.W Gehring Department of Mathematics University of Michigan Ann Arbor, MI 48109 U.SA P.R Halmos Department of Mathematics Santa Clara University Santa Clara, CA 95053 U.SA Mathematical Subject Classifications (1980): 12A35, 12B30, 12C20, 14G20 Library of Congress Cataloging-in-Publication Data Lang,Serge, 1927Cyc\otomic fields l and II (Combined Second Edition)/Serge Lang p cm (Graduate texts in mathematics; 121) Bibliography: p Includes index ISBN 0-387-96671-4 Fields, Aigebraic Cyclotomy QA247.L33 1990 512'.3 dc19 I Title II Series 87-35616 This book is a combined edition of the books previously published as Cyclotomic Fields and Cyclotomic Fields II, by Springer Science+Business Media, LLC, in 1978 and 1980, respectively It contains an additional appendix by Karl Rubin © 1990 by Springer Science+Business Media New York Originally published by Springer-VerlagNew York Inc in 1990 Softcover reprint of the hardcover 2nd edition 1990 AII rights reserved This work may not be translated or copied in whole or in part without the written permission ofthe 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 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 54 ISBN 978-1-4612-6972-4 ISBN 978-1-4612-0987-4 (eBook) DOI 10.1007/978-1-4612-0987-4 Contents Notation Introduction xi xiii CHAPTER Character Sums Character Sums over Finite Fields Stickel berger's Theorem Relations in the Ideal Classes Jacobi Sums as Hecke Characters Gauss Sums over Extension Fields Application to the Fermat Curve 1 14 16 20 22 CHAPTER Stickelberger Ideals and Bernoulli Distributions 26 10 27 32 43 48 The Index of the First Stickel berger Ideal Bernoulli Numbers Integral Stickel berger Ideals General Comments on Indices The Index for k Even The Index for k Odd Twistings and Stickel berger Ideals Stickel berger Elements as Distributions Universal Distributions The Davenport-Hasse Distribution Appendix Distributions 49 50 51 53 57 61 65 v Contents CHAPTER Complex Analytic Class Number Formulas 69 Gauss Sums on Z/mZ Primitive L-series Decomposition of L-series The (± 1)-eigenspaces Cyclotomic Units The Dedekind Determinant Bounds for Class Numbers 72 75 81 84 89 91 69 CHAPTER The p-adic L-function 94 101 Measures and Power Series Operations on Measures and Power Series The Mellin Transform and p-adic L-function Appendix The p-adic Logarithm The p-adic Regulator The Formal Leopoldt Transform The p-adic Leopoldt Transform 95 105 III 112 115 117 CHAPTER Iwasawa Theory and Ideal Class Groups The Iwasawa Algebra Weierstrass Preparation Theorem Modules over Zp[[X]] Zp-extensions and Ideal Class Groups The Maximal p-abelian p-ramified Extension The Galois Group as Module over the Iwasawa Algebra 123 124 129 131 137 143 145 CHAPTER Kummer Theory over Cyclotomic Zp-extensions The Cyclotomic Zp-extension The Maximal p-abelian p-ramified Extension of the Cyclotomic Zp-extension Cyclotomic Units as a Universal Distribution The Iwasawa-Leopoldt Theorem and the Kummer-Vandiver Conjecture 148 148 152 157 160 CHAPTER Iwasawa Theory of Local Units vi The Kummer-Takagi Exponents Projective Limit of the Unit Groups A Basis for U(x) over A The Coates-Wiles Homomorphism The Closure of the Cyclotomic Units 166 166 175 179 182 186 Contents CHAPTER 190 Lubin-Tate Theory 190 196 200 203 204 Lubin-Tate Groups Formal p-adic Multiplication Changing the Prime The Reciprocity Law The Kummer Pairing The Logarithm Application of the Logarithm to the Local Symbol 211 217 CHAPTER 220 Explicit Reciprocity Laws Statement of the Reciprocity Laws The Logarithmic Derivative A Local Pairing with the Logarithmic Derivative The Main Lemma for Highly Divisible x and CI = The Main Theorem for the Symbol xn>n The Main Theorem for Divisible x and CI = unit End of the Proof of the Main Theorems