Math 204B - Number Theory (UC San Diego, winter 2025)

Course description: This is the second in a series of three courses, which is an introduction to algebraic and analytic number theory. Part A treated the basic properties of number fields: their rings of integers, unique factorization and its failure, class numbers, the Dirichlet unit theorem, splitting of primes, cyclotomic fields, and p-adic fields. Part B will consist of an introduction to class field theory (the study of abelian extensions of number fields), together with some exploration of computational tools, particularly SageMath and the LMFDB.

This class will be held in-person (with no livestream), but lectures (including make-up lectures) will be recorded. (You may also find my recordings from winter 2021 relevant.) Office hours will be primarily in person, but I am happy to arrange meetings over Zoom upon request.

This course will use Canvas only to post lecture videos. All other communication will be via email and this website. (You may also message me via the UCSD number theory Zulip; email me to request access.)

Environment: I aim to create a conducive learning environment for those who do not see themselves reflected in the mathematical profession at present and/or have experienced systemic bias affecting their mathematical education. I insist that all participants do their part to maintain this environment. I also aim to address accessibility issues as best I can; please let me know directly if this might affect you.

Instructor: Kiran Kedlaya, kedlaya [at] ucsd [etcetera].

Lectures: MWF 9-9:50am in APM 5402. Recordings will also be posted to Canvas. Some lectures will be cancelled and made up at other times; see below.

Office hours: TBA.

Textbook: No required textbook. For class field theory, I will refer to my own notes on class field theory (available in PDF or HTML). As a supplement I recommend Milne's notes Algebraic Number Theory and Class Field Theory. You may also want to check out Atiyah and MacDonald, Introduction to Commutative Algebra; Lang, Algebraic Number Theory; Fröhlich-Taylor, Algebraic Number Theory; Cassels-Fröhlich, Algebraic Number Theory; Jarvis, Algebraic Number Theory; or Janusz, Algebraic Number Fields.

Prerequisites: Math 204A or permission of instructor. I will grant permission based on background in algebra (at least Math 100A-C, i.e., groups, rings, fields, and Galois theory) and number theory (at the level of Math 104A and 104B). Please do not request enrollment authorization without contacting me separately.

Homework: Weekly problem sets (5 exercises each). There will be eight seven sets in all, released approximately weekly but with no set deadlines. Homework should be submitted by email as a PDF (typed, handwritten on a computer, or photos). You are welcome (and strongly encouraged) to collaborate on homework and/or use online resources, as long as you (a) write all solutions in your own words and (b) cite all sources and collaborators. (However, for best results I recommend trying the problems yourself first.)

Final exam: None. Disregard any information from the UCSD Registrar to the contrary.

Grading: 100% homework. Full credit for any 30 out of the 40 25 out of the 35 total exercises. All work for credit must be submitted by the end of week 10 (March 14).

Key dates:

• First lecture: Monday, January 6.
• No lectures on: Wednesday, January 8; Friday, January 10; Wednesday, January 15; Friday, January 17; Monday, January 20 (university holiday); Monday, February 15 (university holiday).
• Last lecture: Friday, March 7. No lectures during week 10.

Assignments:

• PS 1: pdf.
• PS 2: pdf.
• PS 3: pdf.
• PS 4: pdf.
• PS 5: pdf.

Topics by date (with references):

• Jan 6 (M): The Kronecker-Weber theorem (CFT notes 1.1-1.3).
• Jan 13 (M): The Hilbert class field of a number field (CFT notes 2.1).
• Jan 22 (W): Generalized ideal class groups and the Artin reciprocity law (CFT notes 2.2).
• Jan 24 (F): The existence theorem (CFT notes 2.2); the principal ideal theorem (CFT notes 2.3).
• Jan 24 (F) makeup lecture: Zeta functions and L-functions (CFT notes 2.4).
• Jan 27 (M): Chebotarev density theorem (CFT notes 2.4).
• Jan 29 (W): Cohomology of finite groups I: abstract nonsense (CFT notes 3.1).
• Jan 31 (F): Cohomology of finite groups II: concrete nonsense (CFT notes 3.2).
• Jan 31 (F) makeup lecture: Cohomology of finite groups II: concrete nonsense (CFT notes 3.2).
• Feb 3 (M): Homology and Tate groups (CFT notes 3.3).
• Feb 5 (W): Cohomology of cyclic groups (CFT notes 3.4); extended functoriality (CFT notes 3.2).
• Feb 7 (F): Cohomology of profinite groups (CFT notes 3.5).
• Feb 7 (F) makeup lecture: Cohomology of profinite groups (CFT notes 3.5); overview of local class field theory (CFT notes 4.1).
• Feb 10 (M): Overview of local class field theory (CFT notes 4.1); some computations (CFT notes 4.2).
• Feb 12 (W): Cohomology of local fields: some computations (CFT notes 4.2).
• Feb 14 (F): Local class field theory via Tate's theorem (CFT notes 4.3).
• Feb 14 (F) makeup lecture: Local class field theory via Tate's theorem (CFT notes 4.3).
• Feb 19 (W): Abelian extensions via the Lubin-Tate construction (CFT notes 4.6).
• Feb 21 (F): Adeles (CFT notes 6.1).
• Feb 21 (F) makeup lecture: Ideles and class groups (CFT notes 6.2).
• Feb 24 (M): Adeles and ideles in field extensions (CFT notes 6.3).
• Feb 26 (W): The adelic reciprocity law and Artin reciprocity (CFT notes 6.4).
• Feb 26 (W) makeup lecture: Cohomology of the ideles I: the "First Inequality" (CFT notes 7.1).
• Feb 28 (F): Cohomology of the ideles II: the "Second Inequality" (CFT notes 7.2).
• Feb 28 (F) makeup lecture: Brauer groups and the reciprocity map (CFT notes 7.6).
• Mar 3 (M): The existence theorem (CFT notes 7.5).
• Mar 5 (W): TBA.
• Mar 7 (F): TBA.