Brauer p-divisibility in characteristic p ($p\mathrm{Br}(K)=\mathrm{Br}(K)$)

In characteristic $p$, the exact sequence $1\to K_s^{\times }\stackrel{p}{\to }{K}_s^{\times }\to K_s^{\times }/(K_s^{\times }{)}^p\to 1$ has a quotient killed by $p$. The characteristic p Galois dimension bound makes its second cohomology zero. Hence multiplication by $p$ on $\mathrm{Br}(K)$ is surjective. The root map is injective, and its failure of surjectivity on a separable closure is encoded by the quotient.