Separability

Galois theory for fields sets up a correspondence between the intermediate

fields of a field extension E ⊃ F and subgroups of the group AutF (E) of automorphisms of E over F . To the intermediate field K, E ⊂ K ⊂ F , is associated the subgroup AutK(E), while to the subgroup G is associated the

fixed field EG. It is elementary to check that if an intermediate field K is

the fixed field of any subgroup then it is the fixed field of AutK(E) and if a

subgroup G is the group of automorphisms of E over an intermediate field,

then it is the group AutEG(E). Thus the correspondence sets up a bijection

between intermediate fields which are fixed fields of subgroups and subgroups

which are the full group of automorphisms of E over an intermediate field.