Math 204B - Number Theory

Course description: This is the second in a string of three courses, which is an introduction to algebraic and analytic number theory. Part A treated the basics of number fields (their rings of integers, failure of unique factorization, class numbers, the Dirichlet unit theorem, splitting of primes, cyclotomic fields, and more). In part B, we will focus on local fields and local properties of number fields (completions of number fields, finite extensions of local fields, ramification, different and discriminant, decomposition and inertia groups, basics of local class field theory). Part C will focus on zeta functions. The course will be relatively elementary; no extensive background needed. It should be of interest not just to number theorists, but to algebraic topologists etc. The plan is to cover chapter II (in full) and chapter III (as time permits) in Neukirch's book, and supplement with material from Milne's notes. You should have both.

Instructor: Kiran Kedlaya, kedlaya [at] ucsd [etcetera]. Office hours 2-3:30, or by appointment, in APM 7202.

Lectures: MWF 1-1:50, in APM 7421. There may be also be some makeup lectures to account for cancelled lectures (see announcements).

Textbook: Primarily Algebraic Number Theory (Springer) by J. Neukirch. As a supplement I recommend Milne's notes Algebraic Number Theory and Class Field Theory. You may also want to check out Lang, Algebraic Number Theory; Fröhlich-Taylor, Algebraic Number Theory; Cassels-Fröhlich, Algebraic Number Theory; or Janusz, Algebraic Number Fields. For local fields, the canonical reference is Local Fields I am also working on collating and revising my old class field theory lectures notes from a 2002 course at Berkeley; here is the current draft.

Prerequisites: Math 204A or equivalent (with instructor's permission).

Homework: Weekly problem sets (4-6 exercises), due on Wednesdays.

Final exam: Take-Home. Due Wednesday, March 18, 3 PM; must be submitted by email. (For best results, if you hand-write your exam, please use a scanner rather than the camera on your phone; the department has one available. Of course, you may also submit typed solutions.)

Grading: 50% Homework, 50% Final



Topics covered so far (with references and notes; CFT = class field theory notes, see above):