Exercise 5, § 13

Let K be a field and E an algebraic extension of K of finite degree. a) Suppose that K is of characteristic $p > 0$. Show that if the degree of imperfection of E over K (Ex. 1) is $> 1$, there exist an infinity of distinct fields F such that $K \subset F \subset E$ (reduce to the case where $K(E^p) = K$, and consider the fields $K(a + Xb)$, where the set $\{a, b\}$ is p-free over K and $\lambda \in K$. b) Conclude that for there to exist only a finite number of fields F such that $K \subset F \subset E$ it is necessary and sufficient that the degree of imperfection of E over K be $\leq 1$.