Class Field Theory

by J. Neukirch

Overview

Class field theory, which is so immediately compelling in its main assertions, has, ever since its invention, suffered from the fact that its proofs have required a complicated and, by comparison with the results, rather imper­ spicuous system of arguments which have tended to jump around all over the place. My earlier presentation of the theory [41] has strengthened me in the belief that a highly elaborate mechanism, such as, for example, cohomol­ ogy, might not be adequate for a number-theoretical law admitting a very direct formulation, and that the truth of such a law must be susceptible to a far more immediate insight. I was determined to write the present, new account of class field theory by the discovery that, in fact, both the local and the global reciprocity laws may be subsumed under a purely group­ theoretical principle, admitting an entirely elementary description. This description makes possible a new foundation for the entire theory. The rapid advance to the main theorems of class field theory which results from this approach has made it possible to include in this volume the most important consequences and elaborations, and further related theories, with the excep­ tion of the cohomology version which I have this time excluded. This remains a significant variant, rich in application, but its principal results should be directly obtained from the material treated here.

Product Details

ISBN-13:	9783642824678
Publisher:	Springer Berlin Heidelberg
Publication date:	12/15/2011
Series:	Grundlehren der mathematischen Wissenschaften , #280
Edition description:	Softcover reprint of the original 1st ed. 1986
Pages:	142
Product dimensions:	6.10(w) x 9.25(h) x 0.01(d)

Table of Contents

Chapter I Group and Field Theoretic Foundations 1

1 Infinite Galois Theory 1

2 Profinite Groups 4

3 G-Modules 8

4 The Herbrand Quotient 12

5 Kummer Theory 14

Chapter II General Class Field Theory 18

1 Frobenius Elements and Prime Elements 18

2 The Reciprocity Map 21

3 The General Reciprocity Law 28

4 Class Fields 30

5 Infinite Extensions 32

Chapter III Local Class Field Theory 37

1 The Class Field Axiom 37

2 The Local Reciprocity Law 41

3 Local Class Fields 43

4 The Norm Residue Symbol over Q_p 46

5 The Hilbert Symbol 50

6 Formal Groups 55

7 Fields of πⁿ-th Division Points 60

8 Higher Ramification Groups 64

9 The Weil Group 69

Chapter IV Global Class Field Theory 72

1 Algebraic Number Fields 72

2 Ideles and Idele Classes 76

3 Galois Extensions 81

4 Kummer Extensions 86

5 The Class Field Axiom 89

6 The Global Reciprocity Law 90

7 Global Class Fields 96

8 The Ideal-Theoretic Formulation of Class Field Theory 102

9 The Reciprocity Law of Power Residues 110

Chapter V Zeta Functions and L-Series 113

1 The Riemann Zeta Function 113

2 The Dedekind Zeta Function 117

3 The Dirichlet L-Series 120

4 The Artin L-Series 121

5 The Equality of Dirichlet L-Series and Artin L-Series 128

6 Density Theorems 129

Literature 137

Index 139