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 is a finite Galois extension, the topology of its Galois group is discrete. If the field is the union of finite Galois extensions of a field , the (topological) Galois group is the projective limit of the finite groups where each of the is given the discrete topology, and is a profinite group, hence a totally-disconnected compact topological group. If is the field of invariants for , the subgroup is everywhere dense in . 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 and the subfields of containing .

Comments

Open subgroups of correspond to subfields of that have finite degree over . If is an arbitrary subgroup of , then the extension is Galois (cf. Galois extension), and is the closure of .