Editing Galois connection (section)

====Galois theory====
The motivating example comes from Galois theory: suppose {{math|''L''/''K''}} is a [[field extension]]. Let {{mvar|A}} be the set of all subfields of {{mvar|L}} that contain {{mvar|K}}, ordered by inclusion ⊆. If {{mvar|E}} is  such a subfield, write {{math|Gal(''L''/''E'')}} for the group of [[field automorphism]]s of {{mvar|L}} that hold {{mvar|E}} fixed. Let {{mvar|B}} be the set of subgroups of {{math|Gal(''L''/''K'')}}, ordered by inclusion ⊆. For such a subgroup {{mvar|G}}, define {{math|Fix(''G'')}} to be the field consisting of all elements of {{mvar|L}} that are held fixed by all elements of {{mvar|G}}. Then the maps {{math|''E'' {{mapsto}} Gal(''L''/''E'')}} and {{math|''G'' {{mapsto}} Fix(''G'')}} form an antitone Galois connection.