Algebraic Number Fields and Their Completions

Algebraic Integers

Overview

Z-Orders

Prime and Maximal Ideals

Integral Extensions of Rings

Dedekind Domains

Dedekind Domains

Algebraic Integers in Quadratic Fields

The Chinese Remainder theorem

Fractional Ideals

The Ideal Class Group

Unique Factorization of Ideals in A Dedekind Domain

Localization

Multiplicative Subsets

Semilocal Rings

Discrete Valuation Rings

Ramification Theory

Residue Fields

The Fundamental Identity

Prime Factorization in Quadratic Fields

Ramification, Inertia and Splitting

Prime Factorization in Galois Extensions

The Decomposition Group

The Frobenius Automorphism

Quadratic Example Revisited

p-adic Numbers

Absolute value

Valuations

Topological Equivalence of Absolute Values

p-adic Integers

p-adic Expansions

Hensel's Lemma

Local Fields and Ramification

Local Fields

Absolute Values for Extensions of Local Fields

Integers in Local Fields

Ramification for Local Fields

Producing Totally Ramified Extensions of Local Fields

Prime Factorization in Local Fields

Prime Factorization in Global Fields

Norms

Norms of Elements

Norms of Fractional Ideals

Extensions of Norms

Compatibility of Element and Ideal Norms

An Aside on the Class Group

The Absolute Norm

Connections between Global and Local Extensions

Different and Discriminant

The Different

Finitely Many Primes Ramify

The Discriminant

Discriminant and Ramification

Minkowski Theory

Real and Complex Embeddings

Lattices

Fundamental Domains

Minkowski Lattice Point theorem

Minkowski Bound and Finiteness of the Class Group

Computation of the Class Number

The Unit Group

The Function .I

Roots of Unity

Dirichlet's Unit theorem

Examples

Cyclotomic Fields

Q(~P )

Subfields of Q(~P )

Prime Decomposition in Q(~P )

Q(~P' )

Decomposition of P in Cyclotomic Fields Q(~P' )

Quadratic Reciprocity

Kummer's Lemma

Abelian Extensions of Q

Kronecker-Weber

Problems appear at the end of each chapter.