Cohomological dimension of a field

Remarks

These are the cohomological dimensions $\operatorname{cd}_p(G_K)$ and $\operatorname{cd}(G_K)$ of the profinite group $G_K$. It suffices to test one degree and simple modules: $\operatorname{cd}_p(K) \le n$ if and only if $H^{n+1}(K, M) = 0$ for every simple discrete $G_K$-module $M$ with $pM = 0$. Example: a finite field $\F_q$ has $\operatorname{cd}(\F_q) = 1$, since $G_{\F_q}$ is the profinite completion $\widehat{\Z}$ of $\Z$, whose cohomology with torsion coefficients vanishes above degree $1$, while $H^1(\F_q, \Z/p\Z) = \Hom(\widehat{\Z}, \Z/p\Z) \ne 0$. Example: $\operatorname{cd}(K) = 0$ if and only if $K$ is separably closed. Example: $\R$ has $\operatorname{cd}_2(\R) = \infty$, since $G_\R$ has order $2$ and $H^q(\Z/2\Z, \Z/2\Z) \cong \Z/2\Z$ for all $q \ge 0$. More generally, a field with a nontrivial finite absolute Galois group has infinite cohomological dimension. If $K$ has characteristic $p > 0$, then $\operatorname{cd}_p(K) \le 1$. A finite extension of $\Q_p$ has cohomological dimension $2$.