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

This site will disappear someday, but has been archived here.

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. UCSD offers cross-registration for students from California community colleges and Cal State campuses, and concurrent enrollment for others.

This course will use Canvas only for the gradesheet. All other communication will be via this web site or other tools as described below. (Zoom links will be posted to Canvas and Zulip. If you have not received a Zulip invitation, contact me as soon as possible.)

Online epicourse: 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 means UTC-7 until November 1 and UTC-8 thereafter.)

The epicourse will be entirely unofficial (and free of charge); for credit, one must take the official course (see above). I plan to maintain this arrangement for Math 204B in the winter 2021 quarter. (It may also be possible to do concurrent enrollment for Math 204B if you participate in the epicourse for 204A.)

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. (I will keep updating Zulip even after the course starts, in case you want to join late.)

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 two hours of completion (this is limited by the speed of processing videos).

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; 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; Jarvis, Algebraic Number Theory (the Math 104A/B text); 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) but I plan to be flexible about deadlines. 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.)

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

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

Key dates:

Assignments:

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

  • Oct 2 (F): Overview of the course (video, boards, Miro live). Notes: algebraic numbers and algebraic integers.
  • Oct 5 (M): Gaussian and Eisenstein integers (video, boards, Miro live). References: Neukirch I.1, Jarvis chapter 1, brilliant.org.
  • Oct 7 (W): Eisenstein and other quadratic integers (video, boards, Miro live).
  • Oct 9 (F): Rings of integers in number fields (boards, Miro live). References: Neukirch I.2; Milne chapter 2 (for symmetric polynomials), brilliant.org.
    This lecture was not fully recorded due to a technical issue.
  • Oct 12 (M): Unique factorization of ideals (video, boards, Miro live). References: Neukirch I.2, I.3.
  • Oct 14 (W): Discriminant of a basis, proof of unique factorization, fractional ideals (video/, video, boards, Miro live). References: Neukirch I.2, I.3.
  • Oct 16 (F): The lattice of a number field (video, boards, Miro live). References: Neukirch I.5.
  • Oct 19 (M): Minkowski's theorem (video, boards, Miro live). References: Neukirch I.4, I.5, I.6.
  • Oct 21 (W): The class number; the multiplicative lattice of a number field (video, boards, Miro live). References: Neukirch I.5, I.6, I.7.
  • Oct 23 (F): The multiplicative lattice and the units theorem (video, boards, Miro live). References: Neukirch I.6, I.7.
  • Oct 26 (M): Computational tools for algebraic number theory (video, boards, Miro live).
  • Oct 28 (W): Extensions of Dedekind domains (video, boards, Miro live). References: Neukirch I.8.
  • Oct 30 (F): continuation (video/, boards, Miro live).
  • Nov 2 (M): Cyclotomic fields (video, boards, Miro live). References: Neukirch I.10, Marcus chapter 2.
  • Nov 4 (W): Galois groups, ramification, and splitting (video, boards, Miro live). References: Neukirch I.9.
  • Nov 6 (F): continuation (video, boards, Miro live).
  • Nov 9 (M): Localization (video, boards, Miro live). References: Neukirch I.11.
  • No lecture on Wednesday, November 11.
  • Nov 13 (F): continuation (video, boards, Miro live).
  • Nov 16 (M): Different and discriminant (video, boards, Miro live). References: Neukirch III.2.
  • Nov 18 (W): continuation (video, boards, Miro live).
  • Nov 20 (F): Structure of ramification groups (video, boards, Miro live). References: Neukirch II.10.
  • Nov 23 (M): p-adic numbers (video, boards, Miro live). References: Neukirch II.1.
  • Nov 25 (W): p-adic absolute value (video, boards, Miro live). References: Neukirch II.2, II.4.
  • No lecture on Friday, November 27.
  • Nov 30 (M): Valuations (video, boards, Miro live). References: Neukirch II.3.
  • Dec 2 (W): Extensions of valuations (video, boards, Miro live). References: Neukirch II.4. Notes: extension of valuations.
  • Dec 4 (F): Hensel's lemma (video, boards, Miro live). References: Neukirch II.4.
  • Dec 7 (M): Newton polygons (video, boards, Miro live). References: Neukirch II.6.
  • Dec 9 (W): The Kronecker-Weber theorem: preview of Math 204B (video, boards, Miro live). References: notes on class field theory, chapter 1.
  • Dec 11 (F): The local Kronecker-Weber theorem (video, boards, Miro live). References: notes on class field theory, chapter 1.
  • For fun: This picture is the floor of a shower stall in my house. Why is it relevant here?

    hexagonal tile floor