Syllabus Math 530


Math 530. Algebraic Number Theory Instructor Syllabus

Textbooks used in past semesters:

  • 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.

Topics Covered:

  • Algebraic Background
    Review norm, trace, discriminant, different, integrality, noetherian; Finitely generated torsion-free modules over a PID.
  • Basics
    Number fields, rings of integers being Dedekind domains, integral bases, quadratic and cyclotomic fields.
  • Global theory
    Lattices in Rn, unit theorems, finiteness of class numbers, examples of computing class numbers using Minkowski bound.
  • Local theory
    Completions of Q (and number fields), Hensel's Lemma with application to nonsolvability of Diophantine equations.
  • Decomposition of Primes
    Kummer's Lemma, inverse different, norm of ideals, discriminant, decomposition group, inertia group, Frobenius automorphism, application to quadratic reciprocity.
  • Analytic Methods
    Zeta functions of number fields, Dirichlet L-functions, L(1, c) for quadratic c.