fixed field

Imagine a group of symmetries shuffling the elements of a field. Some elements get moved around; others sit perfectly still no matter which symmetry you apply. The ones that never budge form the fixed field — the 'invariant core' that all the symmetries agree to leave alone.

Formally, let L be a field and H a group of automorphisms of L. The fixed field of H, often written L^H, is the set of all x in L with sigma(x) = x for every sigma in H. This set is closed under addition, multiplication, and inverses (since automorphisms respect those operations), so L^H is a subfield of L containing the prime field. The whole machinery of Galois theory runs on passing back and forth between subgroups H and their fixed fields L^H.

The fixed field reverses inclusions: a bigger group of symmetries imposes more constraints, so it fixes fewer elements, giving a smaller fixed field. In the cleanest setting (Artin's theorem), if H is finite then L is Galois over L^H with Galois group exactly H, and [L : L^H] = |H|. This is the engine that makes the Galois correspondence a genuine bijection rather than a loose pairing.