---
tag: QZMZXFFJFJ
kind: concept
name: Cohomological dimension of a field
msc: [12G10, 12G05]
sources:
  - url: https://en.wikipedia.org/wiki/Cohomological_dimension
---

## Summary

The cohomological dimension of a field $K$ is defined through its [[Galois cohomology|9MPCZ3EAJM]], see the [[definition|XKMBKAN1MR]].
It is the largest degree in which this cohomology with torsion coefficients can be nonzero.
It is the [[cohomological dimension|EF4ZWACHD8]] of the [[absolute Galois group|PGKWBJDTKX]] of $K$, and it can be computed one prime at a time.
Separably closed fields have dimension $0$, finite fields have dimension $1$, and $p$-adic fields have dimension $2$.
It controls which Galois cohomology groups can appear in arithmetic questions over the field.

## Explanations

### L1 human

Galois cohomology attaches to a field a sequence of groups $H^0, H^1, H^2, \ldots$.
For many fields these groups vanish above some degree, and that degree is the cohomological dimension.
A field in which every polynomial splits has nothing above degree $0$.
A finite field has nothing above degree $1$, and the field of $p$-adic numbers has nothing above degree $2$.
For the real numbers the groups never stop, so the dimension is infinite.