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.)

Due to the COVID-19 pandemic, this course will be offered in a hybrid of in-person and remote formats. Lectures will be delivered live when feasible, and otherwise via Zoom; all lectures will be recorded and made available for asynchronous viewing.

At the moment, I can only grant course credit to UCSD enrolled students and others eligible for cross-registration. UCSD also offers "concurrent enrollment" whereby members of the general public can enroll on a per-course basis; I am awaiting clearance to operate concurrent enrollment for this course.

Epicourse: Since lectures will be recorded anyway, I plan to run a parallel "epicourse" for the general public. This will include recorded lectures, some office hours (in various formats), collaborative environments in CoCalc and Zulip, and a mechanism for submitting problem sets for feedback. This will be entirely unofficial; for credit, one must take the official course (see above).

If you would be interested in participating in the epicourse, please fill out this Google Form to help me assess community demand. Keep in mind that all times listed here are local to San Diego: this is UTC-7 until November 1 and UTC-8 thereafter.

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.

Instructor: Kiran Kedlaya, kedlaya [at] ucsd [etcetera]. Office hours: TBA; these will be a mix of in-person and remote (Zoom, text chat).

Lectures: MWF 10-10:50am, in APM B402A. All lectures will be available for remote viewing both synchronously and asynchronously.

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 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 Wednesdays. That said, I plan to be flexible about deadlines. Homework will be submitted online via CoCalc.

Final exam: None.

Grading: 100% homework.

Announcements:

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