Galois topological group

A Galois group endowed with the Krull topology; the filter base (i.e. a basis of the open neighbourhoods of the identity) of this topology consists of normal subgroups of finite index. If $L/K$ is a finite Galois extension, the topology of its Galois group $G(L/K)$ is discrete. If the field $L$ is the union of finite Galois extensions $K_i$ of a field $K$, the (topological) Galois group $G(L/K)$ is the projective limit of the finite groups $G(K_i/K)$ where each of the $G(K_i/K)$ is given the discrete topology, and is a profinite group, hence a totally-disconnected compact topological group. If $K'$ is the field of invariants for $G(L/K)$, the subgroup $G(L/K')$ is everywhere dense in $G(L/K)$. The main theorem on finite Galois extensions may be generalized to infinite extensions: There is a one-to-one correspondence between the closed subgroups of the topological Galois group of a Galois extension $L/K$ and the subfields of $L$ containing $K$.

Comments[edit]

Open subgroups of $G(L/K)$ correspond to subfields of $L$ that have finite degree over $K$. If $H$ is an arbitrary subgroup of $G(L/K)$, then the extension $L/L^H$ is Galois (cf. Galois extension), and $G(L/L^H)$ is the closure of $H$.