Galois cohomology

Summary

Galois cohomology is the group cohomology of Galois groups, see the definition. The main case is the absolute Galois group of a field, acting continuously on a module. Many arithmetic invariants of a field are Galois cohomology groups. Hilbert's Theorem 90 says that $H^1$ of the multiplicative group vanishes. Kummer theory computes $H^1$ of the roots of unity, and $H^2$ of the multiplicative group is the Brauer group. It is a central language of algebraic number theory, class field theory and arithmetic geometry.