Exercise 10, § 15

Let E be an extension of a field K, let G be a compact subgroup of the topological group $\mathrm{Aut}_K(E)$ and let F be the field of invariants of G. Show that E is a Galois extension of F and that G is canonically isomorphic to the topological group $\mathrm{Aut}_F(E)$. (Observe that G operates in a continuous fashion on the discrete space E to establish that an element $x \in E$ has a finite orbit under the action of G.)