Theorems · Theorem · field theory
Algebra.IsIntegral.algebraicIndependent_iff
∀ {ι : Type u_1} (R : Type u_2) {A : Type u_4} {x : ι → A} (S : Type u_5) [inst : CommRing R] [inst_1 : CommRing S]
[inst_2 : CommRing A] [inst_3 : Algebra R S] [inst_4 : Algebra R A] [inst_5 : Algebra S A] [IsScalarTower R S A]
[NoZeroDivisors S] [FaithfulSMul R S] [Algebra.IsIntegral R S], AlgebraicIndependent R x ↔ AlgebraicIndependent S x- Cited by
- 1 results in Mathlib
- Foundations
- Depth 143 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- IsScalarTowerstatement and proof · cited by 3,896
- NoZeroDivisorsstatement and proof · cited by 545
- FaithfulSMulstatement and proof · cited by 340
- Algebra.IsIntegralstatement and proof · cited by 224
- FaithfulSMul.algebraMap_injectiveproof · cited by 198
- AlgebraicIndependentstatement and proof · cited by 120
- AlgebraicIndependent.restrictScalarsproof · cited by 3
- AlgebraicIndependent.extendScalars_of_isIntegralproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- Algebra.IsIntegral.isTranscendenceBasis_iffproof · cited by 0