Theorems · Theorem · field theory
Polynomial.IsRoot.def
∀ {R : Type u} {a : R} [inst : Semiring R] {p : Polynomial R}, p.IsRoot a ↔ Polynomial.eval a p = 0- Defined in
- Mathlib.Algebra.Polynomial.Eval.Defs
- Cited by
- 30 results in Mathlib
- Foundations
- Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Semiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- Polynomialstatement and proof · cited by 5,681
- Polynomial.evalstatement · cited by 796
- Polynomial.IsRootstatement · cited by 152
Cited by30
Results whose statement or proof uses this declaration.
- Polynomial.mem_nthRootsproof · cited by 9
- Polynomial.mul_divByMonic_eq_iff_isRootproof · cited by 5
- IsPrimitiveRoot.minpoly_eq_cyclotomic_of_irreducibleproof · cited by 4
- Polynomial.mem_aroots'proof · cited by 3
- IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_rootsproof · cited by 3
- Polynomial.coprime_of_root_cyclotomicproof · cited by 2
- FiniteField.roots_X_pow_card_sub_Xproof · cited by 2
- Polynomial.X_sub_C_mul_removeFactorproof · cited by 2
- Polynomial.IsWeaklyEisensteinAt.pow_natDegree_le_of_root_of_monic_memproof · cited by 2
- Polynomial.cyclotomic_expand_eq_cyclotomicproof · cited by 2
- Polynomial.cyclotomic_expand_eq_cyclotomic_mulproof · cited by 2
- LinearRecurrence.geom_sol_iff_root_charPolyproof · cited by 2