Math 204A - Number Theory (UCSD and online, fall 2020)

Warning: all material in this syllabus is subject to change until the first lecture. Given the unstable global situation, there may have to be some revisions even after the first lecture, but I will do my best to minimize any disruption.

Course description: This is the first in a series of three courses, which is an introduction to algebraic and analytic number theory. Part A will treat 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 more. There will also be an emphasis on computational tools, particularly SageMath and the LMFDB. (In winter 2021 I will teach Math 204B, which will cover more advanced topics. 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 in-person office hours to the extent possible. (If you were planning on using this course as an in-person course for immigration reasons, contact me for guidance.)

I can only grant course credit to UCSD enrolled students and others eligible for cross-registration. I have not (yet) been cleared to offer "concurrent enrollment" whereby members of the general public can enroll on a per-course basis.

This course will use Canvas only for the gradesheet. All other communication will be via this web site or other tools as described below.

Online epicourse: I plan to run a parallel "epicourse" for the general public. This will include the following components. (Keep in mind that all times are local to San Diego: this is UTC-7 until November 1 and UTC-8 thereafter.)

The epicourse will be entirely unofficial; for credit, one must take the official course (see above). I plan to maintain this arrangement for Math 204B in the winter 2021 quarter.

If you wish to participate in the epicourse, 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. (If you do not receive an invite within 7 days, try emailing me directly.)

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.

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

Textbook: Primarily Algebraic Number Theory (Springer) by J. Neukirch; we will focus on Chapter 1 in this course, and on later chapters in Math 200B. (UCSD affiliates can download the text for free via the UCSD VPN.) As a supplement I recommend Milne's notes Algebraic Number 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; or Janusz, Algebraic Number Fields. Additional references to be added later.

Prerequisites: Math 200A-C (graduate algebra) 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.

Homework: Weekly problem sets (4-6 exercises), due on Thursdays (weeks 2-7 and 9-10). That said, I plan to be flexible about deadlines. Homework will be submitted online via CoCalc.

Final exam: None.

Grading: 100% homework. For full credit, at least 7 of 8 problem sets should be completed in full.

Key dates:

Assignments:

Topics by date (with references, notes, and boards):

For fun: This picture is the floor of a shower stall in my house. Why is it relevant here?

hexagonal tile floor