Exercise 8, § 15

If $E$ is a separable algebraic extension of a field $K$ of characteristic $p > 0$, then the largest perfect subextension $E^{p^\infty}$ of $E$ (the intersection of the fields $E^{p^n}$ for $n \geq 0$) is algebraic over the largest perfect subfield $K^{p^\infty}$ of $K$. (If $x \in E^{p^\infty}$, show that for every integer $n > 0$ we have $x^{p^{-n}} \in K(x)$ and deduce that the minimal polynomial of $x$ over $K^{p^n}$ is the same as its minimal polynomial over $K$.)