CLASS FIELD THEORY EMIL ARTIN JOHN TATE AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island M THE ATI A CA M L 42(4/3,-( N %)3)47 S A O C C I I R E E T !'%7-% Y M A F O 8 U 88 NDED 1 http://dx.doi.org/10.1090/chel/366.H 2000 Mathematics Subject Classification.Primary11R37; Secondary 11–01, 11R34. For additional information and updates on this book, visit www.ams.org/bookpages/chel-366 Library of Congress Cataloging-in-Publication Data Artin, Emil, 1898–1962. Class field theory / Emil Artin, John Tate. Originally published: New York : W. A. Benjamin, 1967. Includes bibliographical references. ISBN 978-0-8218-4426-7 (alk. paper) 1. Class field theory. I. Tate, John Torrence, 1925– joint author. II. Title. QA247.A75 2008 512.7′4—dc22 2008042201 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. 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Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 09 Contents Preface to the New Edition v Preface vii Preliminaries 1 1. Id`eles and Id`ele Classes 1 2. Cohomology 3 3. The Herbrand Quotient 5 4. Local Class Field Theory 8 Chapter V. The First Fundamental Inequality 11 1. Statement of the First Inequality 11 2. First Inequality in Function Fields 11 3. First Inequality in Global Fields 13 4. Consequences of the First Inequality 16 Chapter VI. Second Fundamental Inequality 19 1. Statement and Consequences of the Inequality 19 2. Kummer Theory 21 3. Proof in Kummer Fields of Prime Degree 24 4. Proof in p-extensions 27 5. Infinite Divisibility of the Universal Norms 32 6. Sketch of the Analytic Proof of the Second Inequality 33 Chapter VII. Reciprocity Law 35 1. Introduction 35 2. Reciprocity Law over the Rationals 36 3. Reciprocity Law 41 4. Higher Cohomology Groups in Global Fields 52 Chapter VIII. The Existence Theorem 55 1. Existence and Ramification Theorem 55 2. Number Fields 56 3. Function Fields 59 4. Decomposition Laws and Arithmetic Progressions 62 Chapter IX. Connected Component of Id`ele Classes 65 1. Structure of the Connected Component 65 2. Cohomology of the Connected Component 70 Chapter X. The Grunwald–Wang Theorem 73 1. Interconnection between Local and Global m-th Powers 73 iii iv CONTENTS 2. Abelian Fields with Given Local Behavior 76 3. Cyclic Extensions 81 Chapter XI. Higher Ramification Theory 83 1. Higher Ramification Groups 83 2. Ramification Groups of a Subfield 86 3. The General Residue Class Field 90 4. General Local Class Field Theory 92 5. The Conductor 99 Appendix: Induced Characters 104 Chapter XII. Explicit Reciprocity Laws 109 1. Formalism of the Power Residue Symbol 109 2. Local Analysis 111 3. Computation of the Norm Residue Symbol in Certain Local Kummer Fields 114 4. The Power Reciprocity Law 122 Chapter XIII. Group Extensions 127 1. Homomorphisms of Group Extensions 127 2. Commutators and Transfer in Group Extensions 131 3. The Akizuki–Witt Map v : H2(G, A) H2(G/H, AH )134 4. Splitting Modules and the Principal Ideal→ Theorem 137 Chapter XIV. Abstract Class Field Theory 143 1. Formations 143 2. Field Formations. The Brauer Groups 146 3. Class Formations; Method of Establishing Axioms 150 4. The Main Theorem 154 Exercise 157 5. The Reciprocity Law Isomorphism 158 6. The Abstract Existence Theorem 163 Chapter XV. Weil Groups 167 Bibliography 191 Preface to the New Edition The original preface which follows tells about the history of these notes and the missing chapters. This book is a slightly revised edition. Some footnotes and historical comments have been added in an attempt to compensate for the lack of references and attribution of credit in the original. There are two mathematical additions. One is a sketch of the analytic proof of the second inequality in Chapter VI. The other is several additional pages on Weil groups at the end of Chapter XV. They explain that what is there called a Weil group for a finite Galois extension K/F lacks an essential feature of a Weil group in Weil’s sense, namely the homomorphism ab WK,F Gal(K /F ), but that we recover this once we construct a Weil group for F/F¯ by→ passing to an inverse limit. There is also a sketch of an abstract version of Weil’s proof of the existence and uniqueness of his WK,F for number fields. Ihavenotrenumberedthechapters.Aftersomepreliminaries,thebookstill starts with Chapter V, but the mysterious references to the missing chapters have been eliminated. The book is now in TeX. The handwritten German letters are gone, and many typographical errors have been corrected. I thank Mike Rosen for his help with that effort. For the typos we’ve missed and other mistakes in the text, the AMS maintains a Web page with a list of errata at http://www.ams.org/bookpages/chel-366/ IwouldliketothanktheAMSforrepublishingthisbook,andespeciallySergei Gelfand for his patience and help with the preparation of the manuscript. For those unacquainted with the book, it is a quite complete account of the algebraic (as opposed to analytic) aspects of classical class field theory. The first four chapters, V–VIII, cover the basics of global class field theory, the cohomology of id`ele classes, the reciprocity law and existence theorem, for both number fields and function fields. Chapters IX and X cover two more special topics, the structure and cohomology of the connected component of 1 in the id`ele class group of a number field, and questions of local vs. global behavior surrounding the Grunwald–Wang theorem. Then there are two chapters on higher ramification theory, generalized local classfield theory, and explicit reciprocity laws. This material is beautifully covered also in [21]. For a recent report, see [8]. There is a nice generalization of our classical explicit formula in [13]. The last three chapters of the book cover abstract class field theory. The cohomological algebra behind the reciprocity law is common to both the local and global class field theory of number fields and function fields. Abstracting it led to the definition of a new algebraic structure, ‘class formation’, which embodies the common features of the four theories. The difference is in the proofs that the id`ele classes globally, and the multiplicative groups locally, satisfy the axioms of a class formation. Chapter XIV concludes with a discussion of the reciprocity law and existence theorem for an abstract class formation. In the last v vi PREFACE TO THE NEW EDITION chapter XV, Weil groups are defined for finite layers of an arbitrary class formation, and then, for topological class formations satisfying certain axioms which hold in the classical cases, a Weil group for the whole formation is constructed, by passage to an inverse limit, The class formation can be recovered from its Weil group, and the topological groups which occur as Weil groups are characterized by axioms. The mathematics in this book is the result of a century of developement, roughly 1850–1950. Some history is discussed by Hasse in [5] and in several of the papers in [18]. The high point came in the 1920’s with Takagi’s proof that the finite abelian extensions of a number field are in natural one-to-one correspondence with the quotients of the generalized ideal class groups of that field, and Artin’s proof several years later that an abelian Galois group and the corresponding ideal class group are canonically isomorphic, by an isomorphism which implied all known reciprocity laws. The flavor of this book is strongly influenced by the last steps in that history. Around 1950, the systematic use of the cohomology of groups by Hochschild, Nakayama and the authors shed new light. It enabled many theorems of the local class field theory of the 1930’s to be transferred to the global theory, and led to the notion of class formation embodying the common features of both theo- ries. At about the same time, Weil conceived the idea of Weil groups and proved their existence. With those two developments it is fair to say that the classical one-dimensional abelian class field theory had reached full maturity. There were still a few things to be worked out, such as the local and global duality theories, and the cohomology of algebraic tori, but it was time for new directions. They soon came. For example: Higher dimensional class field theory; • Non-abelian reciprocity laws and the Langlands program; • Iwasawa theory; • Leopold’s conjecture; • Abelian (and non-abelian) ℓ-adic representations; • Lubin-Tate local theory, Hayes explicit theory for function fields, Drinfeld • modules; Stark conjectures; • Serre conjectures (now theorems).