Exercise 3, § 15

Let $K$ be a field of characteristic $p > 0$, $E$ an extension of $K$, and $F$ a finitely generated extension of $E$. Show that the degree of imperfection of $F$ over $K$ is at least equal to that of $E$ over $K$ (reduce to the following two cases : 1" $F$ is separable over $E$; 2" $F = E(x)$, where $x^p \in E$ (cf. V, p. 170, Ex. 2)).