Theorems · Theorem · field theory
transcendental_iff_injective
∀ {R : Type u} {A : Type v} [inst : CommRing R] [inst_1 : Ring A] [inst_2 : Algebra R A] {x : A},
Transcendental R x ↔ Function.Injective ⇑(Polynomial.aeval x)An element x is transcendental over R if and only if the map Polynomial.aeval x
is injective. This is similar to algebraicIndependent_iff_injective_aeval.
- Defined in
- Mathlib.RingTheory.Algebraic.Basic
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 112 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.
- DFunLike.coestatement · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Ringstatement and proof · cited by 7,463
- Polynomialstatement · cited by 5,681
- AlgHomstatement · cited by 3,236
- Polynomial.aevalstatement · cited by 615
- Transcendentalstatement · cited by 91
- Iff.not_leftproof · cited by 49
- isAlgebraic_iff_not_injectiveproof · cited by 2
Cited by6
Results whose statement or proof uses this declaration.
- MvPolynomial.transcendental_supported_polynomial_aeval_Xproof · cited by 4
- algebraicIndependent_unique_type_iffproof · cited by 3
- AlgebraicIndependent.option_iff_transcendentalproof · cited by 3
- Transcendental.infiniteproof · cited by 1
- Algebra.TensorProduct.not_isField_of_transcendentalproof · cited by 1
- transcendental_iff_ker_eq_botproof · cited by 0