Remark

Let $R$ (resp. $S$) be a $p$-radical (resp. separable algebraic) extension of $K$. Then the algebra $R \otimes_K S$ is a field : for $R$ (resp. $S$) is isomorphic to a subextension of $K^{p^\infty}$ (resp. $K_s$) and it suffices to apply the above Cor. and Prop. 1 of $V$, p. 17.