Exercise 3, § 7

Let K be a field of characteristic p > 0, E an extension of finite degree of K, E, and E_s the largest p-radical extension and separable extension of K contained in E. Show that E, \otimes_K E, is isomorphic to the relative separable closure of E, in E, with which it will be identified; if it is not equal to E, then E is an exceptional extension of E, and E, \otimes_K E, is a strongly separable extension of E.

Conversely, let E be an exceptional extension of a field L and let K be a subfield of L such that L is p-radical over K and [L:K] < +\infty; then L is the largest p-radical extension of K contained in E.