文档介绍：Table of Contents
Chapter 1 Introduction
Integral Extensions
Localization
Chapter 2 Norms, Traces and Discriminants
Norms and traces
The Basic Setup For Algebraic Number Theory
The Discriminant
Chapter 3 Dedekind Domains
The Definition and Some Basic Properties
Fractional Ideals
Unique Factorization of Ideals
Some Arithmetic in Dedekind Domains
Chapter 4 Factorization of Prime Ideals in Extensions
Lifting of Prime Ideals
Norms of ideals
A Practical Factorization Theorem
Chapter 5 The Ideal Class Group
Lattices
A Volume Calculation
The Canonical Embedding
Chapter 6 The Dirichlet Unit Theorem
Preliminary Results
Statement and Proof of Dirichlet’s Unit Theorem
Units in Quadratic Fields
1
2
Chapter 7 Cyclotomic Extensions
Some Preliminary Calculations
An Integral Basis of a Cyclotomic Field
Chapter 8 Factorization of Prime Ideals in Galois Extensions
position and Inertia Groups
The Frobenius Automorphism
Applications
Chapter 9 Local Fields
Absolute Values and Discrete Valuations
Absolute Values on the Rationals
Artin-Whaples Approximation Theorem
Hensel’s Lemma
1
A Course In Algebraic Number Theory
Robert B. Ash
Preface
This is a text for a basic course in algebraic number theory, written in accordance with
the following objectives.
1. Provide reasonable coverage for a one-semester course.
2. Assume as prerequisite a standard graduate course in algebra, but cover integral ex-
tensions and localization before beginning algebraic number theory. For general algebraic
background, see my online text “Abstract Algebra: The Basic Graduate Year”, which
can be downloaded from my web site /∼ r-ash/ The abstract algebra
material is referred to in this text as TBGY.
3. Cover the general theory of factorization of ideals in Dedekind domains, as well as the
number field case.
4. Do some detailed calculations i