Proposition 4

— Let $N$ be a Galois extension of $K$. Then the Galois group $\Gamma = \mathrm{Gal}(N/K)$ is compact and totally disconnected.

Every element $\sigma$ of $\Gamma$ has a base of neighbourhoods consisting of the open and closed sets $U_E(\sigma)$, hence $\Gamma$ is totally disconnected (Gen. Top., I, p. 111). We have $\{\sigma\} = \bigcap_{E \in \Lambda} U_E(\sigma)$, hence $\Gamma$ is separated. For each $x \in N$ the set of conjugates $\sigma(x)$ of $x$, as $\sigma$ ranges over $\Gamma$, is finite because $x$ is algebraic over $K$ (V, p. Cor. 1); hence all the projections of $\Gamma$ on the factor spaces of $N^N$ are finite sets, and this shows $\Gamma$ to be relatively compact in $N^N$ (Gen. Top., I, p. 88). It remains to show that $\Gamma$ is closed in $N^N$. Now if $u$ is in the closure of $\Gamma$ in $N^N$, then for each pair of points $(x, y)$ of $N$ there exists $a \in \Gamma$ with $u(x) = \sigma(x), \ u(y) = \sigma(y), \ u(x+y) = \sigma(x+y), \ u(xy) = \sigma(xy)$, whence $u(x+y) = u(x) + u(y)$ and $u(xy) = u(x)u(y)$. By the same reasoning we have $u(x) = x$ for all $x \in K$, hence $u$ is a $K$-homomorphism of $N$ into $N$; since $N$ is algebraic over $K$, $u$ is a $K$-automorphism of $N$ (V, p. 52, Prop. 1), hence $u \in T$.

Let $N$ be a Galois extension of $K$ and $(N_i)_{i \in I}$ an increasing directed family of subextensions of $N$. Suppose that $N_i$ is Galois over $K$ for all $i \in I$ and that $N = \bigcup_{i \in I} N_i$. For each $i \in I$ denote by $\Gamma_i$ the Galois group of $N_i$ over $K$; for $i \leq j$ in $I$ we have $N_i \subset N_j$ and the restriction homomorphism $\varphi_{ij}$ of $\Gamma_j$ into $\Gamma_i$ is defined. It is continuous and so the family $(\Gamma_i, \varphi_{ij})$ is an inverse system of topological groups. Further, for each $i \in I$ denote by $\lambda_i$ the restriction homomorphism of $\mathrm{Gal}(N/K)$ into $\mathrm{Gal}(N_i/K) = \Gamma_i$; it is continuous and we have $\lambda_i = \varphi_{ij} \circ \lambda_j$ for $i \leq j$, hence the family $(\lambda_i)_{i \in I}$ defines a continuous homomorphism $\Lambda$ of $\mathrm{Gal}(N/K)$ into $\varprojlim \Gamma_i$.