Theorems · Theorem · field theory
Polynomial.rootMultiplicity_eq_zero
∀ {R : Type u} [inst : CommRing R] {p : Polynomial R} {x : R}, ¬p.IsRoot x → Polynomial.rootMultiplicity x p = 0- Defined in
- Mathlib.Algebra.Polynomial.Div
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 124 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.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Polynomialstatement and proof · cited by 5,681
- Polynomial.IsRootstatement and proof · cited by 152
- Polynomial.rootMultiplicitystatement · cited by 79
- Polynomial.rootMultiplicity_eq_zero_iffproof · cited by 2
Cited by6
Results whose statement or proof uses this declaration.
- Polynomial.roots_Cproof · cited by 9
- Polynomial.rootMultiplicity_expand_powproof · cited by 2
- Polynomial.rootMultiplicity_sub_one_le_derivative_rootMultiplicityproof · cited by 1
- Polynomial.rootMultiplicity_X_sub_Cproof · cited by 1
- Polynomial.isRoot_of_isRoot_of_dvd_derivative_mulproof · cited by 1
- Polynomial.isRoot_of_isRoot_iff_dvd_derivative_mulproof · cited by 0