Characteristic p Galois dimension bound (${\mathrm{cd}}_p(K)\le 1$)

A field of characteristic $p$ satisfies ${\mathrm{cd}}_p(K)\le 1$. The Artin-Schreier sequence and vanishing of positive-degree cohomology of the additive separable-closure module yield the bound. It concerns discrete $p$-primary modules; it does not assert that the $p$-primary Brauer group vanishes.