Exercise 10, § 14

Let E be an extension of a field K such that the mapping $F \to \mathrm{Aut}_F(E)$ is a bijection of the set of subextensions of E onto the set of subgroups of $\mathrm{Aut}_K(E)$. Show that E is then a Galois extension of K of finite degree. (Using Ex. 9, c), show first that E is necessarily algebraic over K, then use Ex. 9, a.)