---
tag: 9MPCZ3EAJM
kind: concept
name: Galois cohomology
msc: [12G05, 20J06, 11R34]
sources:
  - url: https://en.wikipedia.org/wiki/Galois_cohomology
same_as:
  wikidata: Q382839
  wikipedia: Galois cohomology
---

## Summary

Galois cohomology is the group cohomology of Galois groups, see the [[definition|EESNTXFWHZ]].
The main case is the [[absolute Galois group|PGKWBJDTKX]] of a field, acting continuously on a module.
Many arithmetic invariants of a field are Galois cohomology groups.
[[Hilbert's Theorem 90|YFF8B5A0QG]] says that $H^1$ of the multiplicative group vanishes.
[[Kummer theory|XZQENYV15Y]] computes $H^1$ of the roots of unity, and $H^2$ of the multiplicative group is the [[Brauer group|BEW48TA9WZ]].
It is a central language of algebraic number theory, class field theory and arithmetic geometry.

## Explanations

### L1 human

When a group of symmetries acts on some objects, cohomology measures the ways in which "local" choices fail to fit together into one symmetric choice.
For a field $K$, the symmetries are the automorphisms of a large extension field that fix $K$.
The group $H^0$ is just the set of objects fixed by all symmetries.
The group $H^1$ counts, for example, the different forms over $K$ of an object that become the same over the large field.
For instance, over $\Q$ the field extensions of degree $2$ correspond to the nonzero elements of $H^1$ with coefficients $\set{\pm 1}$.