Synopses & Reviews
Careful organization and clear, detailed proofs characterize this methodical, self-contained exposition of basic results of classical algebraic number theory from a relatively modem point of view. This volume presents most of the number-theoretic prerequisites for a study of either class field theory (as formulated by Artin and Tate) or the contemporary treatment of analytical questions (as found, for example, in Tate's thesis).
Although concerned exclusively with algebraic number fields, this treatment features axiomatic formulations with a considerable range of applications. Modem abstract techniques constitute the primary focus. Topics include introductory materials on elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields.
Subjects correspond to those usually covered in a one-semester, graduate level course in algebraic number theory, making this book ideal either for classroom use or as a stimulating series of exercises for mathematically minded individuals.
Synopsis
Careful organization and clear, detailed proofs make this book ideal either for classroom use or as a stimulating series of exercises for mathematically-minded individuals. Modern abstract techniques focus on introducing elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields.
Synopsis
Ideal either for classroom use or as exercises for mathematically minded individuals, this text introduces elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields.
Table of Contents
Preface; References
Chapter 1. Elementary Valuation Theory
1.1 Valuations and Prime Divisors
1.2 The Approximation Theorem
1.3 Archimedean and Nonarchimedean Prime Divisors
1.4 The Prime Divisors of Q
1.5 Fields with a Discrete Prime Divisor
1.6 e and f
1.7 Completions
1.8 The Theorem of Ostrowski
1.9 Complete Fields with Discrete Prime Divisor; Exercises
Chapter 2. Extension of Valuations
2.1 Uniqueness of Extensions (Complete Case)
2.2 Existence of Extensions (Complete Case)
2.3 Extensions of Discrete Prime Divisors
2.4 Extensions in the General Case
2.5 Consequences; Exercises
Chapter 3. Local Fields
3.1 Newton's Method
3.2 Unramified Extensions
3.3 Totally Ramified Extensions
3.4 Tamely Ramified Extensions
3.5 Inertia Group
3.6 Ramification Groups
3.7 Different and Discriminant; Exercises
Chapter 4. Ordinary Arithmetic Fields
4.1 Axioms and Basic Properties
4.2 Ideals and Divisors
4.3 The Fundamental Theorem of OAFs
4.4 Dedekind Rings
4.5 Over-rings of O
4.6 Class Number
4.7 Mappings of Ideals
4.8 Different and Discriminant
4.9 Factoring Prime Ideals in an Extension Field
4.10 Hilbert Theory; Exercises
Chapter 5. Global Fields
5.1 Global Fields and the Product Formula
5.2 Adèles, Idèles, Divisors, and Ideals
5.3 Unit Theorem and Class Number
5.4 Class Number of an Algebraic Number Field
5.5 Topological Considerations
5.6 Relative Theory; Exercises
Chapter 6. Quadratic Fields
6.1 Integral Basis and Discriminant
6.2 Prime Ideals
6.3 Units
6.4 Class Number
6.5 The Local Situation
6.6 Norm Residue Symbol
Chapter 7. Cyclotomic Fields
7.1 Elementary Facts
7.2 Unramified Primes
7.3 Quadratic Reciprocity Law
7.4 Ramified Primes
7.5 Integral Basis and Discriminant
7.6 Units
7.7 Class Number
Symbols and Notation; Index