Theorems · Theorem · field theory

AlgebraicIndependent.aeval_of_algebraicIndependent

∀ {ι : Type u} {R : Type u_2} {A : Type v} {x : ι → A} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A],
  AlgebraicIndependent R x →
    ∀ {f : ι → MvPolynomial ι R},
      AlgebraicIndependent R f → AlgebraicIndependent R fun i => (MvPolynomial.aeval x) (f i)

If x = {x_i : A | i : ι} and f = {f_i : MvPolynomial ι R | i : ι} are algebraically independent over R, then {f_i(x) | i : ι} is also algebraically independent over R. For the partial converse, see AlgebraicIndependent.of_aeval.

Defined in

Mathlib.RingTheory.AlgebraicIndependent.Basic

Cited by

1 results in Mathlib

Foundations

Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebra