Class Field Theory Book

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