The understanding of the intermediate extensions of a Galois field extension is equivalent to the understanding of its Galois group.
But in general these groups are very large and it is hard to say anything about their structure. Therefore we
limit our attention to abelian intermediate extensions, i.e. those extensions whose Galois group is abelian.
The main topic of this course is to describe the picture for local fields, i.e. finite extensions of the p-adic rational numbers or of the
Laurent field k((t)), where k is a finite field. It turns out that the abelian extensions of a local field K correspond to certain (precisely characterized) subgroups
of the group of units of K. This is a very beautiful result, which is only completely understood since the middle of the 20th century.
In the course we will also explain Galois theory for field extensions of infinite degree and Group cohomology.
These topics are not only needed in the formulation and the proof of the above mentioned result, but nearly in all branches of arithmetic and algebraic geometry.
If time permits we will also discuss how the theory for local fields gives a description of the abelian extensions of number fields (global case).
Requirements
Linear Algebra I, II, Analysis I, II, Algebra I, Number theory (actually only some basic results about local fields are required)