Corollary

— Let $p$ the characteristic exponent of $K$, $\overline{K}$ an algebraic closure of $K$, $K_s$, the relative separable closure of $K$ in $\overline{K}$ and $K^{p^{-\infty}}$ the perfect closure of $K$. Then the canonical homomorphism of $K^{p^\infty} \otimes K_s$ into $\overline{K}$ is an isomorphism.