Theorems · Theorem · field theory

IntermediateField.LinearDisjoint.algEquiv_of_isAlgebraic

∀ {F : Type u} {E : Type v} [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] {A : IntermediateField F E}
  {L : Type w} [inst_3 : Field L] [inst_4 : Algebra F L] [inst_5 : Algebra L E] [inst_6 : IsScalarTower F L E],
  A.LinearDisjoint L →
    ∀ {E' : Type u_1} [inst_7 : Field E'] [inst_8 : Algebra F E'] (B : IntermediateField F E') (L' : Type u_2)
      [inst_9 : Field L'] [inst_10 : Algebra F L'] [inst_11 : Algebra L' E'] [inst_12 : IsScalarTower F L' E']
      (f1 : ↥A ≃ₐ[F] ↥B) (f2 : L ≃ₐ[F] L'), Algebra.IsAlgebraic F ↥A ∨ Algebra.IsAlgebraic F L → B.LinearDisjoint L'

If A and L are linearly disjoint, one of them is algebraic, then for any B and L' isomorphic to A and L respectively, B and L' are also linearly disjoint.

Defined in

Mathlib.FieldTheory.LinearDisjoint

Cited by

0 results in Mathlib

Foundations

Depth 144 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldFieldAlgebraFieldAlgebraAlgebraIsScalarTowerFieldAlgebraFieldAlgebraAlgebraIsScalarTower