Theorems · Theorem · field theory

IsFractionRing.liftAlgHom_fieldRange_eq_of_range_eq

∀ {F : Type u_1} {A : Type u_2} {K : Type u_3} {L : Type u_4} [inst : Field F] [inst_1 : CommRing A]
  [inst_2 : Algebra F A] [inst_3 : Field K] [inst_4 : Algebra F K] [inst_5 : Algebra A K] [inst_6 : IsFractionRing A K]
  [inst_7 : Field L] [inst_8 : Algebra F L] {g : A →ₐ[F] L} [inst_9 : IsScalarTower F A K] (hg : Function.Injective ⇑g)
  {s : Set L}, g.range = Algebra.adjoin F s → (IsFractionRing.liftAlgHom hg).fieldRange = IntermediateField.adjoin F s

The image of IsFractionRing.liftAlgHom is the intermediate field generated by s, if the image of the algebra hom is the subalgebra generated by s.

Defined in

Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra

Cited by

0 results in Mathlib

Foundations

Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldCommRingAlgebraFieldAlgebraAlgebraIsFractionRingFieldAlgebraIsScalarTower