ABSTRACT
The main result here is that inside nice [1] finite-degree field extensions L of k, the intermediate fields K are in (inclusion-reversing) bijection with subgroups H of the Galois group
G = Gal(L/k) = Aut(L/k)
of automorphisms of L over k, by
subgroup H ↔ subfield K fixed by H This is depicted as
G
L | K
H | k
For K the fixed field of subgroup H there is the equality
Further, if H is a normal subgroup of G, then
Gal(K/k) ≈ G/H In the course of proving these things we also elaborate upon the situations in which these ideas apply.
