习题 7, § 14

令 E 为域 K 的扩张和 E 的所有 K-自同构的 $\mathrm{Aut}_K(E)$ 群。 a) 对于 E 在 K 上的每个有限生成子扩展 L，$\mathrm{Aut}_L(E)$ 是 $\mathrm{Aut}_K(E)$ 的子群。 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. 证明，对于 E 在 $K_0$ 上的代数，$\mathrm{Aut}_K(E)$ 是紧的是充分必要的。 e) 假设 K 是无穷大。 如果 $E = K(X)$、域或分数在 K 上不定，则证明 $\mathrm{Aut}_K(E)$ 是无限离散群（V、p. 148，例 11）。 如果K的特征为0，则自同构$\sigma : X \mapsto X + 1$生成的$\mathrm{Aut}_K(E)$的子群$\Gamma$在$\mathrm{Aut}_K(E)$中是闭集，与后者不同，但具有相同的域不变量。