【目录】
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 Idelie Theory
1 Places of a Number Field
2 A Little Topology
3 The Group of Ide1es of a Number Field
4 Cohomology of Finite Cyclic Groups and the Herbrand Quotient
5 Cyclic Galois Action on Ideles
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
l 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 Tare
6 Lubin-Tate Extensions
7 The Local Artin Map
Bibliography
Index
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 Idelie Theory
1 Places of a Number Field
2 A Little Topology
3 The Group of Ide1es of a Number Field
4 Cohomology of Finite Cyclic Groups and the Herbrand Quotient
5 Cyclic Galois Action on Ideles
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
l 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 Tare
6 Lubin-Tate Extensions
7 The Local Artin Map
Bibliography
Index
