Algebraic Number Fields

✍ Scribed by Gerald J. Janusz

• Publisher: Academic Press
• Year: 1973
• Tongue: English
• Leaves: 233
• Series: Pure and Applied Mathematics; 55
• Edition: 1
• Category: Library

✦ Synopsis

The book is directed toward students with a minimal background who want to learn class field theory for number fields. The only prerequisite for reading it is some elementary Galois theory. The first three chapters lay out the necessary background in number fields, such the arithmetic of fields, Dedekind domains, and valuations. The next two chapters discuss class field theory for number fields. The concluding chapter serves as an illustration of the concepts introduced in previous chapters. In particular, some interesting calculations with quadratic fields show the use of the norm residue symbol. For the second edition the author added some new material, expanded many proofs, and corrected errors found in the first edition. The main objective, however, remains the same as it was for the first edition: to give an exposition of the introductory material and the main theorems about class fields of algebraic number fields that would require as little background preparation as possible. Janusz's book can be an excellent textbook for a year-long course in algebraic number theory; the first three chapters would be suitable for a one-semester course. It is also very suitable for independent study.

✦ Table of Contents

Algebraic Number Fields
Copyright Page
Contents
Preface
Chapter I. Subrings of Fields
1. Localization
2. Integral Dependence
3. Discrete Valuation Rings and Dedekind Rings
4. Fractional Ideals and the Class Group
5. Norms and Traces
6. Extensions of Dedekind Rings
7. Discriminant
8. Norms of Ideals
9. Cyclotomic Fields
10. Lattices in Real Vector Spaces
11. The Unit Theorem and Finiteness of the Class Number
Chapter II. Complete Fields
1. Valuations
2. Completions
3. Extensions of Nonarchimedean Valuations
4. Archimedean Valuations
5. Local Norms and Traces and the Product Formula
Chapter III. Decomposition Groups and the Artin Map
1. Decomposition and Inertia Groups
2. The Frobenius Automorphism
3. The Artin Map for Abelian Extensions
Chapter IV. Analytic Methods
1. Moduli and Ray Classes
2. Dirichlet Series
3. Characters of Abelian Groups
4. L-Series and Product Representations
5. Frobenius Density Theorem
Chapter V. Class Field Theory
1. Cohomology of Cyclic Groups
2. Preparations for the Second Inequality
3. A Norm Index Computation
4. The Fundamental Equality for Cyclic Extensions
5. The Reciprocity Theorem
6. Ideal Groups, Conductors, and Class Fields
7. Reduction Steps Toward the Existence Theorem
8. Kummer Extensions and The S-Unit Theorem
9. The Existence Theorem
10. Some Consequences of the Classification Theorem
11. Preliminaries for the Norm Residue Map and the Conductor Theorem
12. Norm Residue Symbol
13. The Hilbert Class Field
Chapter VI. Application of the General Theory to Quadratic Fields

2. Two Examples
3. The Extended Class Group
Appendix A. Normal Basis Theorem and Hilbert’s Theorem 90
Appendix B. Modules over Principal Ideal Domains
Bibliography
Index