A Course in Algebraic Number Theory.pdf

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
SomW Preliminary Calculations
A9 I9tegral Basis of a Cyclotomic Field
Chapter 8 Factorization of Prim6 Ideals in Galois Extensions
positio9 and Inertia Groups
ThW Frobenius Automorphism
Applications
Chapter 9 L7ca5 Fields
AbsolutW Values and DiscretW Valuations
AbsolutW Values o9 thW Rationals
Artin-Whaples Approximatio9 Theorem

Hensel’s Lemma
Chapter 1
Introduction
fi-Chniques of abstract algebra hav- beeW applied to problems iW Wumber theory for a long
time, notably iW th- effort to prov- Fermat’s last theorem. As aW iWtroductory example,
w- will sketC0 a problem for whiC0 aW algebraiC approaC0 works very well. Ifis aW odd
prim- and ∞ 1 mod 4, w- will prov- that is th- sum of two squares, that is, caW
2 2 ∞3 ∈
expressed as +  wher-  and ar- iWtegers. Sinc- 2 is even, it follows that 1
is a quadratiC residu- (that is, a square) mod . fio se- this, pair eaC0 of th- Wumbers
23,..,∈ 2 wit0 its iWvers- mod, and pair 1 wit0 ∈ 1 ≡∈1mod. Th- product of
th- Wumbers 1 throug0∈ 1 is, mod ,
∈ 1 ∈ 1
1 3 2 ×4443 ×∈1 ×∈2 ··43∈
2 2
and therefor-
∈ 1
[( )!]2 ≡∈1modp
2