Theorems · Definition · field theory

IntermediateField.restrictRestrictAlgEquivMapHom

(F : Type u_1) →
  (K : Type u_2) →
    (L : Type u_3) →
      (E : Type u_4) →
        [inst : Field F] →
          [inst_1 : Field K] →
            [inst_2 : Field L] →
              [inst_3 : Field E] →
                [inst_4 : Algebra F K] →
                  [inst_5 : Algebra F L] →
                    [inst_6 : Algebra F E] →
                      [inst_7 : Algebra K E] →
                        [inst_8 : Algebra L E] →
                          [IsScalarTower F K E] → [IsScalarTower F L E] → [Normal F K] → Gal(E/L) →* Gal(K/F)

The map from the Gal(E/L) to Gal(K/F) where E/L/F and E/K/F are two towers of extensions induced by the restriction to K. Note that we do require K/F to be normal but not E/L. If this is the case (and everything is finite dimensional) and K ∩ L = F then this map is surjective, see IntermediateField.restrictRestrictMapHom_surjective. This map is injective if the compositum of K and L is E, see IntermediateField.restrictRestrictAlgEquivMapHom_injective.

Defined in

Mathlib.FieldTheory.Galois.Basic

Cited by

3 results in Mathlib

Foundations

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

Assumes

FieldFieldFieldFieldAlgebraAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerNormal