Math 254A -- Number Theory (UC Berkeley)

Math 254A: Number Theory

Basic information:

  • Faculty: Adam Topaz.
  • Lecture: MWF 11-12P, in 31 Evans.
  • Office Hours: MF 12:10-1:10pm. Office Number: 889 Evans.
  • Syllabus: pdf will be available here soon.
  • Algebraic Number Theory, by S. Lang.
  • Algebraic Number Theory, by J. Neukirch.
  • Algebraic Number Theory, by J.S. Milne (notes freely available online).
  • Algebraic Number Theory, edited by J.W.S. Cassels and A. Fröhlich.
  • … other sources will be discussed in class.

List of Topics:

A strong background in graduate-level algebra, including Galois theory, tensor products, polynomial rings, localization, etc., is required for this course. Here is a (somewhat ambitious) list of the topics I hope to cover in this class, time permitting. This list is tentative and will likely change with the interests of the participants.

  • Integral closure, localization, and other required basics from commutative algebra.
  • Dedekind rings, global fields, rings of integers, factorization of ideals, curves over finite fields, Spec of a commutative ring.
  • Geometry of numbers, class groups, unit groups.
  • Cyclotomic extensions and quadratic extensions.
  • Valuations, ramification and decomposition theory.
  • Completions, local fields, ideles and adeles.
  • Selected topics from (local) class field theory and basics of Galois cohomology.
  • A bit on zeta functions and L-functions, distribution of primes.

Homework

Grading

Final grades will be based on homework (including class participation), and a final project.