Exercise 14, § 10

Let E be an algebraic extension of a field K, $\Gamma$ the group of K-automorphisms of E and $S \subset E$ the field of invariants of $\Gamma$. a) Show that for E to be quasi-Galois over K it is necessary and sufficient for S to be a p-radical extension of K. b) Let $S_s$ be the largest separable extension of K contained in S. Show that $S_s$ is the smallest of the fields F such that $K \subset F \subset E$ and E is a quasi-Galois extension of F. c) Let $E_s$ be the largest separable extension of K contained in E ; show that $E = S(E_s)$ (observe that no K-automorphism of E other than the identity leaves all the elements of $S(E_s)$ invariant).