- Basic Commutative Algebra
- Finitely Generated Abelian Groups
- Noetherian Rings and Modules
- Rings of Algebraic Integers
- Norms and Traces
- Recognizing Algebraic Numbers using Lattice Basis Reduction (LLL)

- Dedekind Domains and Unique Factorization of Ideals

- Factoring Primes

- The Chinese Remainder Theorem

- Discrimants and Norms

- Finiteness of the Class Group

- Dirichlet's Unit Theorem

- Decomposition and Inertia Groups
- Galois Extensions
- Decomposition of Primes:
- The Decomposition Group
- Frobenius Elements
- Galois Representations, -series and a Conjecture of Artin

- Elliptic Curves, Galois Representations, and -functions

- Galois Cohomology

- The Weak Mordell-Weil Theorem

William Stein 2012-09-24