Theorems · Theorem · field theory

IntermediateField.LinearDisjoint.lift_adjoin_rank_eq_lift_rank_right_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 →
    Algebra.IsAlgebraic F ↥A ∨ Algebra.IsAlgebraic F L →
      Cardinal.lift.{w, v} (Module.rank ↥A ↥(IntermediateField.extendScalars ⋯)) =
        Cardinal.lift.{v, w} (Module.rank F L)

If A and L are linearly disjoint over F, one of them is algebraic, then [L(A) : A] = [L : F]. Note that in Lean L(A) is not naturally an A-algebra, so this result is stated in a cumbersome way.

Defined in

Mathlib.FieldTheory.LinearDisjoint

Cited by

4 results in Mathlib

Foundations

Depth 143 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldFieldAlgebraFieldAlgebraAlgebraIsScalarTower