Dover Books on Mathematics

By: Robert B. Ash

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Paperback | 17 June 2010

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128 Pages

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21.59 x 15.75 x 0.64

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This graduate-level text provides coverage for a one-semester course in algebraic number theory. It explores the general theory of factorization of ideals in Dedekind domains as well as the number field case. Detailed calculations illustrate the use of Kummer's theorem on lifting of prime ideals in extension fields.
The author provides sufficient details for students to navigate the intricate proofs of the Dirichlet unit theorem and the Minkowski bounds on element and ideal norms. Additional topics include the factorization of prime ideals in Galois extensions and local as well as global fields, including the Artin-Whaples approximation theorem and Hensel's lemma. The text concludes with three helpful appendixes. Geared toward mathematics majors, this course requires a background in graduate-level algebra and a familiarity with integral extensions and localization.

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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
Table of Contents provided by Ingram. All Rights Reserved.

Dover Books on Mathematics

At a Glance

Paperback

Shipping

How to return your order

Defective items

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