Theorems · Theorem · field theory

Polynomial.Splits.adjoin_rootSet_eq_range

∀ {R : Type u_1} {A : Type u_2} {B : Type u_3} [inst : CommRing R] [inst_1 : Field A] [inst_2 : Algebra R A]
  [inst_3 : CommRing B] [inst_4 : IsDomain B] [inst_5 : Algebra R B] {f : Polynomial R},
  (Polynomial.map (algebraMap R A) f).Splits →
    ∀ (g : A →ₐ[R] B), Algebra.adjoin R (f.rootSet B) = g.range ↔ Algebra.adjoin R (f.rootSet A) = ⊤

Defined in

Mathlib.Algebra.Polynomial.Splits

Cited by

3 results in Mathlib

Foundations

Depth 142 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingFieldAlgebraCommRingIsDomainAlgebra