**Teaching in summer semester 2022 :**

**Local class field theory and Galois cohomology**.

This lecture (4 hrs) + exercises (2 hrs) will be taught in english in the room

**B 252**(Tuesday, Thursday 10:00-12:00) for the lectures and in the room

**B 040**(Tuesday 14:00-16:00) for the exercises.

This lecture will give an introduction to local class field theory and Galois cohomology. We will first introduce and study "local fields", which are certain topological fields, complete and with finite residue field. These are basically obtained by completing "global fields" (that is to say number fields or finite type extensions of transcendence degree 1 of a finite field) at a given discrete valuation. Local class field theory establishes a precise isomorphism of groups between the Galois group G of an abelian finite Galois extension K --> L between local fields and the quotient of the group of units of K modulo the norms of L. To do this we will use Galois cohomology technics and a theorem of Tate. More generally on the way, we will also developp further Galois cohomology and give more examples of applications, for instance to "non-abelian Galois cohomology".

It is recommended to have taken the course "algebraic number field" in the winter semester, although not obligatory. However a good knowledge of the classical theory of number fields and their rings of integers, discrete valutation rings, Dedekind rings, theory of ramification and decomposition of prime ideals in extension of those is required.

**References**:

J.-P. Serre, Corps locaux (french, also available in english under "local fields").

J.-P. Serre, Cohomologie galoisienne (french, also available in english under "Galois cohomology").

S. Lang, algebraic number theory.

J. Neukirch, Klassenkoerpertheorie (german, also available in english under "class field theory")

Exercise sheets .

**Previous semesters .**