Theorems · Theorem · field theory
Polynomial.le_rootMultiplicity_iff
∀ {R : Type u} [inst : CommRing R] {p : Polynomial R},
p ≠ 0 → ∀ {a : R} {n : ℕ}, n ≤ Polynomial.rootMultiplicity a p ↔ (Polynomial.X - Polynomial.C a) ^ n ∣ pThe multiplicity of a as root of a nonzero polynomial p is at least n iff
(X - a) ^ n divides p.
- Defined in
- Mathlib.Algebra.Polynomial.Div
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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
- RingHomstatement · cited by 10,189
- Polynomialstatement and proof · cited by 5,681
- Polynomial.Xstatement and proof · cited by 1,639
- Polynomial.Cstatement and proof · cited by 1,598
- pow_zeroproof · cited by 1,094
- Dvd.dvd.transproof · cited by 148
- Nat.findproof · cited by 139
- Polynomial.rootMultiplicitystatement · cited by 79
- pow_dvd_powproof · cited by 56
- one_dvdproof · cited by 37
Cited by10
Results whose statement or proof uses this declaration.
- Polynomial.rootMultiplicity_le_iffproof · cited by 4
- Polynomial.rootMultiplicity_mul_X_sub_C_powproof · cited by 3
- Polynomial.count_map_rootsproof · cited by 2
- Polynomial.rootMultiplicity_eq_natTrailingDegree'proof · cited by 1
- Polynomial.eq_rootMultiplicity_mapproof · cited by 1
- Polynomial.le_rootMultiplicity_mapproof · cited by 1
- Polynomial.le_rootMultiplicity_mulproof · cited by 1
- Multiset.prod_X_sub_C_dvd_iff_le_rootsproof · cited by 0
- Polynomial.rootMultiplicity_addproof · cited by 0