Theorems · Theorem · field theory

IntermediateField.map_fixingSubgroup

∀ {F : Type u_1} {E : Type u_2} (E' : Type u_3) [inst : Field F] [inst_1 : Field E] [inst_2 : Field E']
  [inst_3 : Algebra F E] [inst_4 : Algebra F E'] [inst_5 : Algebra E E'] [inst_6 : IsScalarTower F E E']
  (L : IntermediateField F E) [inst_7 : Normal F E],
  (IntermediateField.map (IsScalarTower.toAlgHom F E E') L).fixingSubgroup =
    Subgroup.comap (AlgEquiv.restrictNormalHom E) L.fixingSubgroup

If K / E / k is a field extension tower with E / k normal, L is an intermediate field of E / k, then the fixing subgroup of L viewed as an intermediate field of K / k is equal to the preimage of the fixing subgroup of L viewed as an intermediate field of E / k under the natural map Aut(K / k) → Aut(E / k) (AlgEquiv.restrictNormalHom).

Defined in

Mathlib.FieldTheory.Galois.Basic

Cited by

1 results in Mathlib

Foundations

Depth 152 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldFieldFieldAlgebraAlgebraAlgebraIsScalarTowerNormal