English
The Galois correspondence forms a Galois insertion between the dual of fixingSubgroup and fixedField, giving a tight relation between fields and subgroups.
Русский
Галуа–соответствие образует Galois–défi между двойником fixingSubgroup и fixedField, образуя строгую связь между полями и подгруппами.
LaTeX
$$IsGalois galoisInsertionIntermediateFieldSubgroup$$
Lean4
/-- The Galois correspondence as a `GaloisInsertion`. -/
def galoisInsertionIntermediateFieldSubgroup [FiniteDimensional F E] :
GaloisInsertion
(OrderDual.toDual ∘ (IntermediateField.fixingSubgroup : IntermediateField F E → Subgroup (E ≃ₐ[F] E)))
((IntermediateField.fixedField : Subgroup (E ≃ₐ[F] E) → IntermediateField F E) ∘ OrderDual.toDual)
where
choice K _ := IntermediateField.fixingSubgroup K
gc K H := (IntermediateField.le_iff_le H K).symm
le_l_u H := le_of_eq (IntermediateField.fixingSubgroup_fixedField H).symm
choice_eq _ _ := rfl
