Theorems · Theorem · field theory
Polynomial.mul_divByMonic_eq_iff_isRoot
∀ {R : Type u} {a : R} [inst : CommRing R] {p : Polynomial R},
(Polynomial.X - Polynomial.C a) * (p /ₘ (Polynomial.X - Polynomial.C a)) = p ↔ p.IsRoot a- Defined in
- Mathlib.Algebra.Polynomial.Div
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 120 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.
Cites20
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
- zero_addproof · cited by 2,366
- Polynomial.Xstatement and proof · cited by 1,639
- MulZeroClass.zero_mulproof · cited by 1,625
- Polynomial.Cstatement and proof · cited by 1,598
- sub_selfproof · cited by 996
- Polynomial.evalproof · cited by 796
- Polynomial.eval_Xproof · cited by 172
- Polynomial.IsRootstatement · cited by 152
Cited by5
Results whose statement or proof uses this declaration.
- Polynomial.dvd_iff_isRootproof · cited by 15
- minpolyDiv_specproof · cited by 7
- Polynomial.X_sub_C_mul_removeFactorproof · cited by 2
- Polynomial.mul_div_eq_iff_isRootproof · cited by 2
- Polynomial.Splits.of_natDegree_eq_twoproof · cited by 1