Course description: This is the second in a string of three courses, which is an introduction to algebraic and analytic number theory. In part II we will discuss the basics of local fields. We will start from scratch with absolute values on fields, completions, Ostrowski, Hensel's lemma, ramification, Witt vectors, and then move on to division algebras over local fields, Brauer groups, Hasse invariant, and local class field theory. If time permits we will discuss formal groups and Lubin-Tate theory.
Instructor: Claus Sorensen, csorensen [at] ucsd [etcetera]. Office hours by appointment in APM 6151.
Lectures: MWF 11-11:50, in APM 2402. (Beginning Mon Jan 7th.)
Textbooks/notes:[Neu] J. Neukirch, Algebraic Number Theory, Springer Berlin Heidelberg, 1999. (E-version available here.)
[Lor] F. Lorenz, Algebra Volume II: Fields with structure, algebras and advanced topics (Book) Springer, 2008 (Universitext).
[Ser] J.-P. Serre, Local fields (Book) Springer, 1979 (GTM).
[ANT] J. Milne, Algebraic Number Theory, Notes, v. 2018 (Univ. of Michigan)
[CFT] J. Milne, Class Field Theory, Notes, v. 2018 (Univ. of Michigan)
Prerequisites: Math 204A
Homework: Almost weekly problem sets (9 altogether, ~3 exercises each) posted below, due Wed in class; cf. the calendar.
Final exam: Take-Home. Due Monday, March 18th, 3 PM. Posted HERE
Grading: 60% Homework, 40% Final (you must take the final to pass the course).
Assignments: