A Short Course on Galois Cohomology - William Stein - University of ...

2 Introduction
Number Theory is the study of G = Gal( ¯ Q/Q), the group of automorphisms
of the algebraic closure ¯ Q, and the sets G naturally acts on. The following
question, for example, is an open problem: is every finite group a quotient
of G?
Galois cohomology involves studying the group G by applying homological
algebra. This provides a natural way to classify objects, e.g. twists
of a curve, and linearizes problems by defining new invariants, revealing
previously hidden structure.
This course will consists of mainly two parts, one on group cohomology
in greater generality and one on Galois cohomology. In the first part, we
will apply homological algebra to groups, solving problems like the following:
given a group G acting on an abelian group A, find all extensions of G by
A, that is, exact sequences
0 → A → E → G → 1,
where we write both 0 and 1 for the trivial group with one element. In the
second part, we will apply group cohomology to number theory. This is a
very important tool, vital to most advances in algebraic number theory in
the last thirty years.
We now give further details of the course structure. The first part will
cover the basic theory, involving essentially no number theory:
• G-modules
• cocycles, coboundaries
• basic homological algebra
• dimension shifting
• inflation/ restriction
• cup products, etc.
We will include most proofs in this part. In the second part, on Galois
cohomology, we will specialize to number theory, focussing more on examples
and including fewer proofs. The topics we will consider are:
• Profinite groups and their cohomology; topological groups
• Hilbert 90, and applications
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