Math 204B - Number Theory (UCSD and online, winter 2021)

This site will disappear someday, but has been archived here.

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 continue with these topics, plus an introduction to class field theory (the study of abelian extensions of number fields). As in part A, there will be some emphasis on computational tools, particularly SageMath and the LMFDB. (In spring 2021, Claus Sorensen will teach Math 204C.)

Due to the COVID-19 pandemic and UCSD campus regulations, this course will be offered in a fully remote format. Lectures will be delivered live via Zoom, and also recorded for asynchronous viewing. Office hours will be held via Zoom; I also plan to offer some in-person office hours as conditions permit.

I can only grant course credit to UCSD enrolled students. UCSD offers cross-registration for students from California community colleges and Cal State campuses, and concurrent enrollment for others (for a fee which I do not control).

This course will use Canvas in the following ways only.

Online epicourse: As with 204A, I plan to run a parallel "epicourse" for the general public. This will include the following components. (All times are local to San Diego, which is UTC-8.)

The epicourse will be entirely unofficial (and free of charge); for credit, one must take the official course (see above).

If you wish to participate in the epicourse and did not participate in fall 2020, please fill out this Google Form. You should receive an invitation to join Zulip; this may take a day or two depending on how often I check the form. (I will keep updating Zulip even after the course starts, in case you want to join late.) If you participated in the epicourse in fall 2020, you do not need to sign up again.

Note that the lecture recordings, whiteboards, and lecture notes will be posted publicly here, so you do not need to join the epicourse to get those. However, I hope the other interactions via Zoom and Zulip will add significant value, and encourage everyone following the lectures to join.

Environment: In both the course and the epicourse, 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 10-10:50am, via Zoom (meeting code 964 2065 5406). All lectures will be available for remote viewing both synchronously and asynchronously. I aim to have each lecture posted within one hour of completion (this is limited by the speed of video processing).

Office hours: Unless otherwise specified, these are for both the UCSD course and the epicourse. Timings may be adjusted during the term.

Textbook: Primarily Algebraic Number Theory (Springer) by J. Neukirch. (UCSD affiliates can download the text for free via the UCSD VPN.) For class field theory, I will also 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 (the Math 104A/B text); or Janusz, Algebraic Number Fields. Additional references to be added later.

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 as they were taught in 2019-2020). Please do not request enrollment authorization without contacting me separately.

If you were not following Math 204A and are depending on your independent knowledge of that material, I would recommend viewing the last two lectures (December 9 and 11) of that course in advance. They serve as an introduction to the material we will be considering here.

Homework: Weekly problem sets (4-7 exercises), due on Thursdays (weeks 2-10). Homework will be submitted online via CoCalc. 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.)

The maximum homework score will be 5 (formerly 6) out of 9 complete homeworks (by percentage, combining partial scores across all submitted assignments.) I am also offering the option to make up any one problem set by submitting a short writeup of a topic that is related to the class but not covered in lecture. Please contact me if you wish to exercise this option.

Since the homework policy is more generous than last quarter, I am going to be a bit less generous about extensions this term; please ask for one in advance of the due date if you need it. This will allow me to open up homework discussion on Zulip in a more timely fashion.

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

Grading: 100% homework; see above.

Key dates:

Assignments: The numbering continues that of 204A.

Topics by date (with videos, references, notes, and boards): See also this page for the videos embedded as iframes, or this playlist for all the videos together.