Algebraic Number Theory II (Class Field Theory)

Lectures

Monday, 2 - 4 pm, room WSC S-U-4.02

Wednesday, 2 - 4 pm, room WSC S-U-3.03

Problem session

Tuesday, 2 - 4 pm, room WSC S-U-3.03

Prerequisites

The lecture is independent of Algebraic Number Theory I. However, I do expect the participants to be well acquainted with the techniques provided by an introductory course on abstract algebra. Depending on the audience the lecture will be given in English.

Contents

Algebraic number theory studies the field of rational numbers and its finite field extensions, the so-called algebraic number fields. Completing an algebraic number field at a fixed absolute value yields what is known as a local field, e.g. the real or the p-adic numbers.

Class field theory deals with the problem of classifying all abelian extensions of a local field or an algebraic number field, i.e. all Galois extensions with abelian Galois groups. Initially, this problem was solved by Hasse and Artin and is of great importance up to the present day.

The course will follow the cohomological approach to class field theory introduced by Tate. In particular, it will provide a thorough introduction to the cohomology of finite groups. This is a technique which is of importance in many fields of mathematics.

Further topics include: a reminder of Galois theory, structure theory of local fields, cohomology of groups, class formations, local class field theory and the local reciprocity law, overview of global class field theory

References

The course will roughly follow the first sections of [2] in the following list of titles.

[1] S. Lang: Algebraic Number Theory; Graduate Texts in Mathematics 110, 2nd Edition, Springer, 1994

[2] J. Neukirch: Class Field Theory; edited by A. Schmidt, online edition, Springer, 2012

[3] J. Neukirch: Algebraische Zahlentheorie; Springer, 1992

Credits

In order to be given credit points for this course you will need to take an oral exam in the end. In order to be admitted to the exam it is necessary to obtain at least 45% of the points on the problem sheets and to actively participate in the problem sessions.

Problem sheets

The problem sheets will be distributed and collected on Tuesday during the problem sessions.