Math 620 - Algebraic Number Theory

MWF 1:00pm - 1:50pm, Room: Math 0405

Lecturer: Larry Washington (For email address, hold cursor here and look at the bottom of the page)
Office: Math 4415, Phone: 301-405-5116
Office Hours: I'm in my office most of the time.
Syllabus:
When working in number theory, one is led quickly to the study of rings of algebraic integers in finite extensions of the rationals, for example, the Gaussian integers. The study of these rings, and applications, will occupy the first half of the course. Analytic techniques such as zeta functions will then be introduced. Each of the last four topics listed below will be treated briefly, if time permits.
  • Topics from ring theory, especially Dedekind domains, prime ideals, unique factorization, and ramification.
  • Finiteness of the class number and the Dirichlet Unit Theorem
  • Cyclotomic reciprocity and quadratic reciprocity
  • Zeta functions, $L$-series, and Dirichlet's theorem on primes in arithmetic progressions
  • $p$-adic numbers
  • Equations over finite fields
  • Elliptic curves and other diophantine equations
  • Modular forms with applications to sums of squares.
    Prerequisite: Math 601, in particular some ring theory and Galois theory. Text: There is a free, online text that is close to what I'll do in my lectures: Algebraic Number Theory notes by James Milne
    Another good online source is Matt Baker's notes
    Homework: Homework 1
    Homework 2
    Homework 3