Theorems · Theorem · field theory
Polynomial.comp_eq_zero_iff
∀ {R : Type u} [inst : Semiring R] [NoZeroDivisors R] {p q : Polynomial R},
p.comp q = 0 ↔ p = 0 ∨ Polynomial.eval (q.coeff 0) p = 0 ∧ q = Polynomial.C (q.coeff 0)- Defined in
- Mathlib.Algebra.Polynomial.Degree.Lemmas
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 118 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringNoZeroDivisors
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
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
- Semiringstatement and proof · cited by 13,802
- RingHomstatement · cited by 10,189
- Polynomialstatement and proof · cited by 5,681
- Polynomial.Cstatement and proof · cited by 1,598
- Polynomial.natDegreeproof · cited by 1,105
- Polynomial.coeffstatement and proof · cited by 1,045
- Polynomial.evalstatement and proof · cited by 796
- NoZeroDivisorsstatement and proof · cited by 545
- Polynomial.compstatement and proof · cited by 193
- mul_eq_zeroproof · cited by 94
- Polynomial.C_0proof · cited by 40
Cited by3
Results whose statement or proof uses this declaration.
- Polynomial.Gal.splits_in_splittingField_of_compproof · cited by 1
- Polynomial.div_tendsto_atBot_zero_iff_degree_ltproof · cited by 0
- Polynomial.abs_div_tendsto_atBot_atTop_of_degree_gtproof · cited by 0