= Brauer p-divisibility in characteristic p
{c}
{title2=$p\operatorname{Br}(K)=\operatorname{Br}(K)$}
In characteristic $p$, the exact sequence $1\to K_s^{\times}\xrightarrow{p}K_s^{\times}\to K_s^{\times}/(K_s^{\times})^p\to1$ has a quotient killed by $p$. The <characteristic p Galois dimension bound> makes its second cohomology zero. Hence multiplication by $p$ on $\operatorname{Br}(K)$ is surjective. The root map is injective, and its failure of surjectivity on a separable closure is encoded by the quotient.
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