Exercise 7, § 14

Let E be an extension of a field K and $\mathrm{Aut}_K(E)$ the group of all K-automorphisms of E. a) For every finitely generated subextension L of E over K, $\mathrm{Aut}_L(E)$ is a subgroup of $\mathrm{Aut}_K(E)$. Show that when L ranges over the set of subextensions of E finitely generated over K, the groups $\mathrm{Aut}_L(E)$ form a fundamental system of neighbourhoods of the neutral element of $\mathrm{Aut}_K(E)$ for a topology compatible with the group structure of $\mathrm{Aut}_K(E)$ and for which $\mathrm{Aut}_K(E)$ is a separated and totally disconnected group. b) When E is equipped with the discrete topology, the topology of $\mathrm{Aut}_K(E)$ is the coarsest in which the mapping $(u, x) \to u(x)$ of $\mathrm{Aut}_K(E) \times E$ into E is continuous. \* c) The group $\mathrm{Aut}_\mathbf{Q}(\mathbf{R})$ is reduced to the neutral element (V, p. 153, Ex. 2 of § 9). d) Let $K_0 \supset K$ be the subextension of E consisting of the elements invariant under all K-automorphisms of E. Show that for E to be algebraic over $K_0$ it is necessary and sufficient that $\mathrm{Aut}_K(E)$ should be compact. e) Assume that K is infinite. If $E = K(X)$, the field of fractions in one indeterminate over K, show that $\mathrm{Aut}_K(E)$ is an infinite discrete group (V, p. 148, Ex. 11). If K is of characteristic 0, the subgroup $\Gamma$ of $\mathrm{Aut}_K(E)$ generated by the automorphism $\sigma : X \mapsto X + 1$ is closed in $\mathrm{Aut}_K(E)$ and distinct from the latter, but has the same field of invariants.