Proposition 5

— The homomorphism $\lambda$ of $\mathrm{Gal}(N/K)$ into $\varprojlim \mathrm{Gal}(N_i/K)$ is an isomorphism of topological groups.

Since $\mathrm{Gal}(N/K)$ is compact, $\lambda$ continuous and the group $\varprojlim \mathrm{Gal}(N_i/K)$ separated, it suffices to show that $h$ is bijective (Gen. Top., I, p. 87, Cor. 2). Let $u = (u_i)_{i \in I}$ be an element of $\varprojlim \mathrm{Gal}(N_i/K)$; for each $i \in I$, $u_i$ is a K-automorphism of $N_i$ and $u_i$ is the restriction of $u_j$ to $N_i$ for $i \leq j$. Since $N = \bigcup_{i \in I} N_i$, there exists a unique element $\sigma$ of $\mathrm{Gal}(N/K)$ which coincides with $u_i$ on $N_i$ for all $i \in I$. Thus $\sigma$ is the unique element of $\mathrm{Gal}(N/K)$ such that $\lambda(u) = u$, hence $\lambda$ is bijective.

This applies in particular when we take for the family $(N_i)$ the family of all finite Galois subextensions of $N$; then each group $\mathrm{Gal}(N_i/K)$ is discrete and finite. The topological group $\mathrm{Gal}(N/K)$ is thus isomorphic to a directed inverse limit of finite groups, equipped with the discrete topology; this is sometimes called a profinite topological group.