Theorems · Theorem · field theory
Polynomial.mem_roots
∀ {R : Type u} {a : R} [inst : CommRing R] [inst_1 : IsDomain R] {p : Polynomial R}, p ≠ 0 → (a ∈ p.roots ↔ p.IsRoot a)- Defined in
- Mathlib.Algebra.Polynomial.Roots
- Cited by
- 30 results in Mathlib
- Foundations
- Depth 133 from the axioms · uses propext, Classical.choice, Quot.sound
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.
- CommRingstatement and proof · cited by 17,173
- Polynomialstatement and proof · cited by 5,681
- Multisetstatement · cited by 2,627
- IsDomainstatement and proof · cited by 2,196
- Polynomial.rootsstatement · cited by 264
- Polynomial.IsRootstatement · cited by 152
- Polynomial.mem_roots'proof · cited by 8
Cited by30
Results whose statement or proof uses this declaration.
- Polynomial.mem_nthRootsproof · cited by 9
- Polynomial.finite_setOfPred_isRootproof · cited by 7
- IsPrimitiveRoot.isRoot_cyclotomicproof · cited by 4
- isRoot_of_unity_iffproof · cited by 3
- Polynomial.card_roots_toFinset_le_card_roots_derivative_sdiff_roots_succproof · cited by 3
- FiniteField.roots_X_pow_card_sub_Xproof · cited by 2
- Polynomial.mem_roots_mapproof · cited by 2
- Polynomial.mem_roots_map_of_injectiveproof · cited by 2
- Algebra.IsAlgebraic.lift_cardinalMk_le_sigma_polynomialproof · cited by 2
- Polynomial.zero_of_eval_zeroproof · cited by 2
- Polynomial.roots_eq_of_degree_le_card_of_ne_zeroproof · cited by 2
- IsAlgClosed.roots_eq_zero_iffproof · cited by 1