Theorems · Theorem · field theory

IsTranscendenceBasis.nonempty_iff_transcendental

∀ {ι : Type u} {R : Type u_1} {A : Type w} {x : ι → A} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A]
  [Nontrivial R], IsTranscendenceBasis R x → (Nonempty ι ↔ Algebra.Transcendental R A)

If x is a transcendence basis of A/R, then it is not empty if and only if A/R is transcendental.

Defined in

Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis

Cited by

1 results in Mathlib

Foundations

Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraNontrivial