Local Fields Seminar — SS 2019
Organization
 Teachers

Prof.Dr. A. Huber
and Johan Commelin
 Location
 SR 404, ErnstZermeloStraße 1
 Time
 Wednesday, 8ct.
Description
The real numbers form a completion of the rational numbers,
and other completions are given by the socalled $p$adic numbers.
These are the first examples of local fields.
Local fields are a very important concept in the study
of number fields (finite extensions of $\mathbb{Q}$),
because they allow us to study problems "locally".
For example, one of the main goals in number theory
is to study solutions of polynomial equations over the integers or the rationals.
This is a very hard problem, but one can make some progress by studying
the solutions locally over the $p$adic numbers for every prime $p$.
In this seminar we will follow the book "Local Fields" by Serre,
and explore the basic properties of local fields.
The goals of this seminar are a proof of the local Kronecker–Weber theorem
and the statement of local class field theory.
At the end of this seminar, students should be well prepared
to study the proof of (local) class field theory,
one of the highlights of number theory in the previous century.
Schedule
The program can be found
here.
 Date  Name  Title 

1 
24.04 
KL 
Discrete valuation rings 
1 
08.05 
KL 
Discrete valuation rings, II 
2 
15.05 
DB 
Extensions 
3 
22.05 
tba 
Topology 
4 
29.05 
SK 
Witt vectors 
5 
05.06 
BP 
Structure of complete DVRs 
6 
19.06 
tba 
Discriminants and Differents 
7 
26.06 
FG 
Higher ramification groups 
8 
03.07 
tba 
The norm map 
9 
10.07 
MH 
The Hasse–Arf theorem 
11 
17.07 
CS 
Local KroneckerWeber 
12 
24.07 
MB 
Statement of local class field theory 
Literature
The main source is the book by Serre,
Local Fields.
Another book that we will refer to is Algebraïsche Zahlentheorie
by Neukirch.