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 consist of an introduction to class field theory (the study of abelian extensions of number fields), together with some exploration of computational tools, particularly SageMath and the LMFDB.
This class will be held in-person (with no livestream), but lectures will be recorded. Office hours will primarily in person (hybrid upon request), occasionally over Zoom.
This course will use Canvas in the following ways only.
Environment: 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 9-9:50am, room TBA. Recordings will also be posted to Canvas. Some lectures will be cancelled and made up at other times.
Office hours: TBA.
Textbook: No required textbook. For class field theory, I will 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; or Janusz, Algebraic Number Fields.
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). Please do not request enrollment authorization without contacting me separately.
Homework: TBA.
Final exam: None. Disregard any information from the UCSD Registrar to the contrary.
Grading: 100% homework; see above.
Key dates:
Assignments: TBA.
Topics by date (with references):