# Number Theory III: Class Field Theory

**Time**: Tuesday 12:00 -- 14:00**Locatio**n: A6/SR 031**First Lecture**: 16.10.2018

**Übungen**: Friday, 08:00 -- 10:00, A6/SR 032

## Course Description

### Reference

- J.S. Milne,
*Class Field Theory*(v4.02), 2013. Available at https://www.jmilne.org/math/CourseNotes/CFT.pdf.

### Goal

- Chapter II : cohomology of groups and of profinite groups, on the way revise (profinite) Galois Theory.
- Chapter III: Local Class Field Theory in Characteristic 0.

## Addendum

- Oct. 30th, 2018 (updated)
- Nov 27th, 2018
- CAUTION: no class on Tuesday Dec. 4th, 2018 due to the birthday of the Freie Universität.
- On Friday Dec. 7th, 2018, during the exercise class (8h15-9h45), there will instead a lecture on Tate's theorem (Milne Thm 3.11) and related topics. There will be no correction of exercises that week, Exercise Sheet 7 will be treated the week after on Dec. 14th, 2018.

**On Friday Jan. 18th, Jan. 25th and Feb. 1st, during the exercise class (8h15-9h45), there will be instead supplementary lectures providing additional material and context on local class field theory. There will be no more exercise sheets.**- Jan 15th, 2019

## Exam

Here are the two mathematical texts on which the exam is based:

- Serre's "Local fields" Chapter IX section 4. This corresponds to p.142-143 in the following scan of the chapter.
- Part of Milne's "Class field theory" Chapter IV section 4 (available online, see above), more specifically from the bottom of p.136 from "A non archimedean local field" to p.138 before Rmk 4.4.

The goal is to return on Feb. 8th, before 10:00am, in Simon’s mailbox two documents explaining the statements and proofs of the texts. **The page limit is of 3 pages for each document.**

When you need references to other statements, either from the course or from the context, write very precisely what is needed.

You can write the documents in LaTeX if you wish. Please separate into two different documents the two texts, so they can be corrected separately.

The deadline is strict, there will be no exception. The limitation on the number of pages is strict as well.

## Klausureinsicht

The Klausureinsicht will take place on Thursday February 14th at 12h30.