类域论

1. A Brief Review

1. Number Fields

2. Completions of Number Fields

3. Some General Questions Motivating Class Field Theory

2. Dirichlefs Theorem on Primes in Arithmetic Progressions

1. Characters of Finite Abelian Groups

2. Dirichlet Characters

3. Dirichlet Series

4. Dirichlet抯 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 Idelic Theory

1. Places of a Number Field

2. A Little Topology

3. The Group of Id^ies 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 Preibmnary 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

《类域论（英文版）》将gauss、legendre和其他的二次和更高阶的互反率巧妙结合，并将这些结果更加一般化，是学习类域理论的入门书籍。《类域论（英文版）》运用传统方法和原始技巧呈现书中的材料，思路清晰流畅，是这个领域的图书很难企及的。《类域论（英文版）》可以作为代数数论的研究生教程，尤其适合自学。书中有大量的练习贯穿始终，已经被证明了是一本很成功的教材。