Theorems · Theorem · commutative algebra
Polynomial.exists_monic_aeval_eq_zero_forall_mem_pow_of_mem_map
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
[Algebra.IsIntegral R S] {I : Ideal R} {x : S},
x ∈ Ideal.map (algebraMap R S) I →
∃ p, p.Monic ∧ (Polynomial.aeval x) p = 0 ∧ ∀ (i : ℕ), p.coeff i ∈ I ^ (p.natDegree - i)- Cited by
- 1 results in Mathlib
- Foundations
- Depth 135 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites42
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- SetLike.coeproof · cited by 8,199
- Set.ofPredproof · cited by 6,101
- Polynomialstatement and proof · cited by 5,681
- Set.imageproof · cited by 5,609
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- AlgHomstatement · cited by 3,236
- mul_assocproof · cited by 1,667
Cited by1
Results whose statement or proof uses this declaration.
- Polynomial.exists_monic_aeval_eq_zero_forall_mem_of_mem_mapproof · cited by 0