Theorems · Theorem · field theory
IsAlgClosed.dvd_iff_roots_le_roots
∀ {k : Type u} [inst : Field k] [IsAlgClosed k] {p q : Polynomial k}, p ≠ 0 → q ≠ 0 → (p ∣ q ↔ p.roots ≤ q.roots)- Defined in
- Mathlib.FieldTheory.IsAlgClosed.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 140 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldIsAlgClosed
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- Multisetstatement · cited by 2,627
- Polynomial.rootsstatement · cited by 264
- IsAlgClosedstatement and proof · cited by 150
- IsAlgClosed.splitsproof · cited by 29
- Polynomial.Splits.dvd_iff_roots_le_rootsproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- Polynomial.isRoot_of_isRoot_iff_dvd_derivative_mulproof · cited by 0
- IsAlgClosed.associated_iff_roots_eq_rootsproof · cited by 0