Franz Lemmermeyer
# Class Field Theory

Seminar: Class Field Theory a la Hasse. English translation of Hasse's
`Vorlesungen über Klassenkörpertheorie. My intention is to
cover Part I: Galois Theory in this semester.

What should you expect? First of all a very explicit introduction to
Galois theory (the Lectures are from 1932, today's standard version of
Galois theory was created in Artin's Notre Dame lectures in the
1950s). Actually Hasse deals only with fields of characteristic 0,
thus avoiding the problems with inseparable extensions. He then
discusses applications of Galois theory to algebraic number theory
(Hilbert's theory of ramification, connections with the local theory
of finite extensions of the p-adic numbers), and finally introduces
the Artin symbol.

Prerequisites: abstract algebra (homomorphisms, factor groups).

## Schedule

We will meet Wednesday 3:40 except on days when there's a
general seminar.
- W 25.02.04, 3:40 First meeting; outline; history.
A few comments.
- W 03.03.04, 3:40 The Galois group
- W 10.03.04, 3:40 The Galois group of Q(i,2
^{1/4})/Q
- W 17.03.04, 3:40 The Main Theorem of Galois Theory
- W 24.03.04 No seminar
- W 31.03.04 3:40 Ramification groups: definition and examples
- W 07.04.04 3:40 Ramification groups: simple properties
- W 14.04.04 General seminar
- W 21.04.04 No seminar
- W 28.04.04 3:40 Ramification groups
- May: Artin symbol, statements of class field theory

## Notes

Here's the first chapter covering
Galois theory proper, Hilbert's ramification groups, and the Artin
symbol. If you observe any typos or strange formulations, please
let me know. Here are a few
problems we may discuss.