Galois connection between subgroups and subfields

Summary

Let $L/K$ be a field extension. One map sends a group $H$ of automorphisms of $L$ over $K$ to its fixed field. The other sends an intermediate field $E$ to the group of automorphisms that fix $E$. Together they form an antitone Galois connection. Both maps reverse inclusions, and applying one after the other can only enlarge a subgroup or an intermediate field. For a finite Galois extension the two maps are inverse bijections; this is the fundamental theorem of Galois theory.