Theorems · Theorem · field theory
Polynomial.modByMonic_eq_zero_iff_dvd
∀ {R : Type u} [inst : Ring R] {p q : Polynomial R}, q.Monic → (p %ₘ q = 0 ↔ q ∣ p)- Defined in
- Mathlib.Algebra.Polynomial.Div
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Ring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites27
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Ringstatement and proof · cited by 7,463
- Polynomialstatement and proof · cited by 5,681
- one_mulproof · cited by 2,841
- Nontrivialproof · cited by 2,416
- zero_addproof · cited by 2,366
- MulZeroClass.mul_zeroproof · cited by 2,091
- WithBotproof · cited by 1,498
- Polynomial.natDegreeproof · cited by 1,105
- Polynomial.degreeproof · cited by 643
- Polynomial.leadingCoeffproof · cited by 498
- Polynomial.Monicstatement and proof · cited by 461
- mul_subproof · cited by 201
Cited by16
Results whose statement or proof uses this declaration.
- minpoly.dvdproof · cited by 31
- Polynomial.dvd_iff_isRootproof · cited by 15
- minpoly.isIntegrallyClosed_dvdproof · cited by 9
- Polynomial.map_dvd_mapproof · cited by 5
- minpoly.unique'proof · cited by 4
- Polynomial.pow_mul_divByMonic_rootMultiplicity_eqproof · cited by 3
- Polynomial.generalizedEisensteinproof · cited by 1
- Polynomial.mem_ker_modByMonicproof · cited by 1
- Polynomial.modByMonic_oneproof · cited by 1
- Polynomial.dvd_of_infinite_eval_dvd_evalproof · cited by 0
- Polynomial.isCoprime_of_is_root_of_eval_derivative_ne_zeroproof · cited by 0
- Polynomial.mul_self_modByMonicproof · cited by 0