Cohomological dimension of a field

Summary

The cohomological dimension of a field $K$ is defined through its Galois cohomology, see the definition. It is the largest degree in which this cohomology with torsion coefficients can be nonzero. It is the cohomological dimension of the absolute Galois group 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.