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1 A Brief Review 1

1 Number Fields 1

2 Completions of Number Fields 8

3 Some General Questions Motivating Class Field Theory 14

2 Dirichlet's Theorem on Primes in Arithmetic Progressions 17

1 Characters of Finite Abelian Groups 17

2 Dirichlet Characters 20

3 Dirichlet Series 30

4 Dirichlet's Theorem on Primes in Arithmetic Progressions 35

5 Dirichlet Density 40

3 Ray Class Groups 45

1 The Approximation Theorem and Infinite Primes 45

2 Ray Class Groups and the Universal Norm Index Inequality 47

3 The Main Theorems of Class Field Theory 60

4 The Idèlic Theory 63

1 Places of a Number Field 64

2 A Little Topology 66

3 The Group of Idèles of a Number Field 68

4 Cohomology of Finite Cyclic Groups and the Herbrand Quotient 75

5 Cyclic Galois Action on Idèles 83

5 Artin Reciprocity 105

1 The Conductor of an Abelian Extension of Number Fields and the Artin Symbol 105

2 Artin Reciprocity 111

3 An Example:Quadratic Reciprocity 128

4 Some Preliminary Results about the Artin Map on Local Fields 130

6 The Existence Theorem,Consequences and Applications 135

1 The Ordering Theorem and the Reduction Lemma 136

2 Kummer n-extensions and the Proof of the Existence Theorem 139

3 The Artin Map on Local Fields 148

4 The Hilbert Class Field 153

5 Arbitrary Finite Extensions of Number Fields 159

6 Infinite Extensions and an Alternate Proof of the Existence Theorem 162

7 An Example:Cyclotomic Fields 168

7 Local Class Field Theory 181

1 Some Preliminary Facts About Local Fields 182

2 A Fundamental Exact Sequence 186

3 Local Units Modulo Norms 191

4 One-Dimensional Formal Group Laws 195

5 The Formal Group Laws of Lubin and Tate 198

6 Lubin-Tate Extensions 201

7 The Local Artin Map 210

Bibliography 219

Index 223