Theorems · Theorem · field theory
Irreducible.dvd_iff_aeval_eq_zero
∀ {A : Type u_1} {B : Type u_2} [inst : Field A] [inst_1 : Ring B] [inst_2 : Algebra A B] [Nontrivial B]
{p q : Polynomial A}, Irreducible p → ∀ {b : B}, (Polynomial.aeval b) p = 0 → ((Polynomial.aeval b) q = 0 ↔ p ∣ q)- Defined in
- Mathlib.FieldTheory.Minpoly.Field
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 123 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldRingAlgebraNontrivial
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
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
- Algebrastatement and proof · cited by 11,388
- Ringstatement and proof · cited by 7,463
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- AlgHomstatement · cited by 3,236
- Nontrivialstatement and proof · cited by 2,416
- MulZeroClass.zero_mulproof · cited by 1,625
- Polynomial.Cproof · cited by 1,598
- map_mulproof · cited by 1,137
- Polynomial.aevalstatement and proof · cited by 615
- Polynomial.leadingCoeffproof · cited by 498
Cited by1
Results whose statement or proof uses this declaration.
- Irreducible.natDegree_dvd_iff_dvd_X_pow_card_pow_sub_Xproof · cited by 0