Math 204B - Number Theory

Course description: This is the second in a sequence of three courses, which together constitute 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), adeles and ideles, and the basics of local and global class field theory. Part C will focus on zeta functions and L-functions.

Instructor: Kiran Kedlaya, kedlaya [at] ucsd [etcetera]. Office hours: Th 11-12, in APM 7202.

Lectures: MWF 10-10:50, in APM 7421.

Textbook: No required text. Recommended: Algebraic Number Theory (Springer) by J. Neukirch. If you do not already have a copy, I suggest downloading the PDF version from any UCSD computer. Some additional references you might find helpful (but don't try to read them all!):

• Some lecture notes of mine on class field theory.
• Milne's notes Algebraic Number Theory and Class Field Theory.
• Lang, Algebraic Number Theory.
• Fröhlich-Taylor, Algebraic Number Theory.
• Cassels-Fröhlich, Algebraic Number Theory.
• Janusz, Algebraic Number Fields.
• Serre, Local Fields.

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

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

Final exam: None.

Grading: 100% homework.

Announcements:

• First lecture: Monday, January 9.
• Holidays this term: Monday, January 16; Monday, February 20.
• No lecture Monday, March 13.
• Last problem set due Friday, March 17 (not Wednesday).

Assignments:

• HW 1 (Due Jan 18): pdf.
• HW 2 (Due Feb 1): pdf (updated Jan 30).
• HW 3 (Due Feb 15): pdf.
• HW 4 (Due Mar 1): pdf.
• HW 5 (Due Mar 17): pdf.

Topics covered so far:

• Jan 9: overview; the field of p-adic numbers via algebraic constructions (Neukirch II.1).
• Jan 11: the field of p-adic numbers as a metric completion (Neukirch II.2).
• Jan 13 (lecture by Claus Sorensen): absolute values on fields, completions, archimedean vs. nonarchimedean, equivalence of absolute values, Ostrowski's theorem for Q (Neukirch II.3).
• Jan 18: approximation theorem (Neukirch II.3); completions (Neukirch II.4).
• Jan 20: Ostrowski's theorem on archimedean valuations; more on discrete valuations; filtration on units (Neukirch II.4).
• Jan 23 (lecture by Alina Bucur): Hensel's lemma; extension of valuations (Neukirch II.4).
• Jan 25 (lecture by Claus Sorensen): local fields (Neukirch II.5).
• Jan 27 (lecture by Cristian Popescu): structure of the multiplicative group of a local field (Neukirch II.5).
• Jan 30: Newton polygons (Neukirch II.6).
• Feb 1: more on Newton polygons; slope factorizations of polynomials over a henselian field (Neukirch II.6).
• Feb 3: fundamental identity (Neukirch II.6); unramified extensions (Neukirch II.7).
• Feb 6: classification of unramified extensions; tamely ramified extensions (Neukirch II.7).
• Feb 8: cyclotomic extensions and their ramification (Neukirch II.7); extensions of valuations and embeddings (Neukirch II.8).
• Feb 10: extensions of valuations, the fundamental identity (Neukirch II.8); decomposition, inertia, ramification groups (Neukirch II.9).
• Feb 13: comparison of decomposition groups and local Galois groups (Neukirch II.9).
• Feb 15: higher ramification groups (Neukirch II.10).
• Feb 17: higher ramification groups continued.
• Feb 22: higher ramification groups continued.
• Feb 24: discriminant and different, Artin conductor (Neukirch III.2, VI.11).
• Feb 27: conductor-discriminant formula (Neukirch VI.11).
• Mar 1: class field theory over Q: the Kronecker-Weber theorem (CFT 1, 3).
• Mar 3: Hilbert class field, Artin reciprocity law (CFT 4, 5).
• Mar 6: Principal ideal theorem, local reciprocity law (CFT 6, 12).
• Mar 8: Local existence theorem, norm limitation theorem, comments on H^1 and H^2 (CFT 9, 12).
• Mar 10: Adeles and ideles (CFT 16).
• Mar 13 (lecture by Alireza Salehi Golsefidy): Adeles and ideles in field extensions, adelic reciprocity, the First Inequality (CFT 17, 18, 19).
• Mar 15: no lecture
• Mar 17: Analytic methods and the Second Inequality; Cheboratev density theorem; preview of L-functions (CFT 7, 19).