Math 530. Algebraic Number Theory

Number Fields, Marcus, 1987, Springer-Verlag, NY

A Classical Introduction to Modern Number Theory, M. Rosen and K. Ireland, GTMmath #84, 1982, Springer

Algebraic Number Fields, Janusz, 1996, Amer. Math. Soc.

1. Algebraic Background
   Review norm, trace, discriminant, different, integrality, noetherian; Finitely generated torsion-free modules over a PID.

2. Basics
   Number fields, rings of integers being Dedekind domains, integral bases, quadratic and cyclotomic fields.

3. Global theory
   Lattices in Rⁿ, unit theorems, finiteness of class numbers, examples of computing class numbers using Minkowski bound.

4. Local theory
   Completions of Q (and number fields), Hensel's Lemma with application to nonsolvability of Diophantine equations.

5. Decomposition of Primes
   Kummer's Lemma, inverse different, norm of ideals, discriminant, decomposition group, inertia group, Frobenius automorphism, application to quadratic reciprocity.

6. Analytic Methods
   Zeta functions of number fields, Dirichlet L-functions, L(1, c) for quadratic c.