Artin's theorem on fixed fields

Artin's theorem is the quiet workhorse that makes Galois theory rest on solid ground. It says that a finite group of symmetries of a field is never wasteful: it cuts the field down by exactly its own size and no more. This single quantitative fact is what turns the loose pairing of groups and fields into an exact, reversible correspondence.

Precisely: if L is a field and G is a finite group of automorphisms of L, with fixed field K = L^G, then L/K is a finite Galois extension, the degree [L : K] equals the order |G|, and the full automorphism group Gal(L/K) is precisely G. Artin proved this by a clean linear-algebra argument, showing the elements of G are linearly independent as functions on L (the independence of characters) and bounding [L : K] above by |G|, then matching it from below.

The strategic value is that it lets you build Galois extensions from groups rather than from polynomials: pick any finite group of automorphisms, and its fixed field is automatically a base over which L is Galois with that exact group. This is the converse direction of the fundamental theorem and the reason every subgroup of a Galois group really does arise as Gal(L/F) for the right F — without Artin's theorem the Galois correspondence could fail to be surjective onto subgroups.