Course description: This is the first in a series of three courses, which together constitute 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.
While this course is being conceived as an in-person course and will be taught as such (to the extent possible), all aspects of the course will accommodate remote participation. Lectures will be recorded for asynchronous viewing. Office hours will be held both in person and via Zoom. Asynchronous communication will take place over Zulip (see below). Homework will be submitted online using CoCalc (see below).
As I did when I taught this course in fall 2020, I plan to make the course content available to the general public as a sort of "epicourse". Use this Google Form to sign up.
Environment: I commit to creating a conducive learning environment for all students, but with extra attention to those who do not see themselves reflected in the mathematical profession at present or have experienced systemic bias affecting their mathematical education. I insist that all participants do their part to maintain this environment. Please do not hesitate to bring any concerns to me, particularly regarding accessibility issues that I may not have anticipated; I am always looking for ways to improve.
Instructor: Kiran Kedlaya, kedlaya [at] ucsd [etcetera].
Lectures: MWF 11-11:50am, APM B412 (starting Monday, Sep 26). A few lectures will be given remotely via Zoom. Recordings of all lectures will be posted via Zulip.
Office hours: Wed 2-3pm, hybrid (in-person in APM 7202 or Zoom); Wed 5-6pm, remote (over Zoom).
Textbook: Primarily Algebraic Number Theory (Springer) by J. Neukirch; we will focus on Chapter 1 in this course, with a bit of material from Chapter 2. (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 (this text has been used for Math 104A/B); or Janusz, Algebraic Number Fields.
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 Math 104B).
Homework: Weekly problem sets (5 exercises each), due on Thursdays (weeks 2-8 and 10) but I will be somewhat flexible about dates. 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. Full credit for any 30 out of the 40 total exercises.
Topics by date (with references): See also my fall 2020 course page.