fundamental theorem of Galois theory

Inside a Galois extension there are usually many fields stacked between the base field and the top — and inside its Galois group there are many subgroups. The fundamental theorem says these two collections are perfect mirror images of each other: every intermediate field matches exactly one subgroup, and vice versa, in a tidy reversible dictionary.

Precisely, for a finite Galois extension L/K with Galois group G, there is a one-to-one, order-reversing correspondence between the fields F with K ⊆ F ⊆ L and the subgroups H of G. To a field F you assign the subgroup of automorphisms fixing F; to a subgroup H you assign its fixed field. Bigger fields match smaller subgroups, which is what “order-reversing” means.

The dictionary carries extra structure too: the degree of L over an intermediate field equals the size of the matching subgroup, and an intermediate field is itself a normal (Galois) extension of K exactly when its matching subgroup is a normal subgroup of G — in which case the smaller Galois group is a quotient group. This is the engine behind nearly every application of Galois theory.