# Local Fields Seminar — SS 2019

## Organization

Teachers
Prof.Dr. A. Huber and Johan Commelin
Location
SR 404, Ernst-Zermelo-Straße 1
Time
Wednesday, 8ct.

## Description

The real numbers form a completion of the rational numbers, and other completions are given by the so-called $p$-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 Kronecker--Weber 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.