1 A Brief Review

1 Number Fields

2 Completions of Number Fields

3 Some General Questions Motivating Class Field Theory

2 Dirichlet's Theorem on Primes in Arithmetic Progressions

1 Characters of Finite Abelian Groups

2 Dirichlet Characters

3 Dirichlet Series

4 Dirichlet's Theorem on Primes in Arithmetic Progressions

5 Dirichlet Density

3 Ray Class Groups

1 The Approximation Theorem and Infinite Primes

2 Ray Class Groups and the Universal Norm Index Inequality

3 The Main Theorems of Class Field Theory

4 The Idèlic Theory

1 Places of a Number Field

2 A Little Topology

3 The Group of Idèles of a Number Field

4 Cohomology of Finite Cyclic Groups and the Herbrand Quotient

5 Cyclic Galois Action on Idèles

5 Artin Reciprocity

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

2 Artin Reciprocity

3 An Example:Quadratic Reciprocity

4 Some Preliminary Results about the Artin Map on Local Fields

6 The Existence Theorem,Consequences and Applications

1 The Ordering Theorem and the Reduction Lemma

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

3 The Artin Map on Local Fields

4 The Hilbert Class Field

5 Arbitrary Finite Extensions of Number Fields

6 Infinite Extensions and an Alternate Proof of the Existence Theorem

7 An Example:Cyclotomic Fields

7 Local Class Field Theory

1 Some Preliminary Facts About Local Fields

2 A Fundamental Exact Sequence

3 Local Units Modulo Norms

4 One-Dimensional Formal Group Laws

5 The Formal Group Laws of Lubin and Tate

6 Lubin-Tate Extensions

7 The Local Artin Map

Bibliography

Index