Exercise 4, § 16

Let K be a field of characteristic $p > 0$ whose absolute degree of imperfection (V, p. 170, Ex. 1) is equal to 1. Show that for an extension E of K to be separable over K it is necessary and sufficient that E should contain no element p-radical over K and not belonging to K (if $a \in K$ forms ap-basis of K over $K^p$ show that $K^{p^{-1}} = K(a^{p^{-1}})$ and E are linearly disjoint over K).