just take every subgroup we can in general find two different subgroups that fix the same intermediate field. Therefore we amend this by introducing a topology on the Galois group. Let be a Galois extension (possible infinite) and let be the Galois group of the extension. Let be the set of the Galois groups of all finite intermediate Galois extension. Note that for all we can define the maps by . We then define the Krull Topology on to be weakest topology such that for all the maps are continuous, where we endow each with the discrete topology. Stated differently as an inverse limit of topological groups (where again each is endowed with the discrete topology). This makes a profinite group (in fact every profinite group can be realised as the Galois group of a Galois extension, see for example ). Note that when is finite, the Krull topology is the discrete topology. Now that we have defined a topology on the Galois group we can restate the fundamental theorem for infinite Galois extensions. Let denote the set of all finite intermediate field extensions of and let denote the set of all closed subgroups of endowed with the Krull topology. Then there exists a bijection between and given by the map defined by and the map defined by . One important thing one need to check is that is a well-defined map, that is that is a closed subgroup of for all intermediate. For a proof see for example. See also Galois connection References External links Theorems in group theory Galois theory