The motivating example comes from Galois theory: suppose is a field extension. Let be the set of all subfields of that contain , ordered by inclusion ⊆. If is such a subfield, write for the group of field automorphisms of that hold fixed. Let be the set of subgroups of , ordered by inclusion ⊆. For such a subgroup , define to be the field consisting of all elements of that are held fixed by all elements of . Then the maps and form an antitone Galois connection.