Theorems · Theorem · field theory
AlgebraicIndependent.isTranscendenceBasis_iff_isAlgebraic
∀ {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],
AlgebraicIndependent R x → (IsTranscendenceBasis R x ↔ Algebra.IsAlgebraic (↥(Algebra.adjoin R (Set.range x))) A)- Cited by
- 3 results in Mathlib
- Foundations
- Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Set.Elemproof · cited by 7,166
- Set.ofPredproof · cited by 6,101
- Set.rangestatement and proof · cited by 4,705
- Nontrivialstatement and proof · cited by 2,416
- Subalgebrastatement · cited by 1,353
- Algebra.adjoinstatement and proof · cited by 535
- LE.le.antisymmproof · cited by 507
- Algebra.IsAlgebraicstatement and proof · cited by 322
- AlgebraicIndependentstatement and proof · cited by 120
Cited by3
Results whose statement or proof uses this declaration.
- isTranscendenceBasis_iff_algebraicIndependent_isAlgebraicproof · cited by 1
- IsTranscendenceBasis.sumElim_compproof · cited by 1
- IsTranscendenceBasis.algebraMap_compproof · cited by 0