A Course in Algebraic Number Theory

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ISBN: 9780486477541 | 0486477541
Cover: Paperback
Copyright: 6/17/2010

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This text for a graduate-level course covers the general theory of factorization of ideals in Dedekind domains as well as the number field case. It illustrates the use of Kummer's theorem, proofs of the Dirichlet unit theorem, and Minkowski bounds on element and ideal norms. 2003 edition.

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Introduction	p. 1
Integral Extensions	p. 2
Localization	p. 5
Norms, Traces and Discriminants	p. 9
Norms and traces	p. 9
The Basic Setup for Algebraic Number Theory	p. 13
The Discriminant	p. 16
Dedekind Domains	p. 21
The Definition and Some Basic Properties	p. 21
Fractional Ideals	p. 22
Unique Factorization of Ideals	p. 25
Some Arithmetic in Dedekind Domains	p. 27
Factorization of Prime Ideals in Extensions	p. 31
Lifting of Prime Ideals	p. 31
Norms of ideals	p. 34
A Practical Factorization Theorem	p. 37
The Ideal Class Group	p. 41
Lattices	p. 41
A Volume Calculation	p. 43
The Canonical Embedding	p. 44
The Dirichlet Unit Theorem	p. 49
Preliminary Results	p. 49
Statement and Proof of Dirichlet's Unit Theorem	p. 52
Units in Quadratic Fields	p. 54
Cyclotomic Extensions	p. 57
Some Preliminary Calculations	p. 57
An Integral Basis of a Cyclotomic Field	p. 60
Factorization of Prime Ideals in Galois Extensions	p. 65
Decomposition and Inertia Groups	p. 65
The Frobenius Automorphism	p. 68
Applications	p. 70
Local Fields	p. 73
Absolute Values and Discrete Valuations	p. 73
Absolute Values on the Rationals	p. 76
Artin-Whaples Approximation Theorem	p. 77
Completions	p. 79
Hensel's Lemma	p. 82
Appendices	p. 85
Quadratic Reciproicty Via Gauss Sums	p. 85
Extension of Absolute Values	p. 92
The Different	p. 94
Solutions to Problems	p. 97
Index	p. 109
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A Course in Algebraic Number Theory

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