Theorems · Theorem · field theory
Polynomial.Monic.not_irreducible_iff_exists_add_mul_eq_coeff
∀ {R : Type u} [inst : CommSemiring R] [NoZeroDivisors R] {p : Polynomial R},
p.Monic → p.natDegree = 2 → (¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂)- Defined in
- Mathlib.Algebra.Polynomial.Monic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringNoZeroDivisors
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites35
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommSemiringstatement and proof · cited by 10,911
- Polynomialstatement and proof · cited by 5,681
- Finset.sumproof · cited by 5,195
- Nat.cast_oneproof · cited by 2,501
- Nontrivialproof · cited by 2,416
- zero_addproof · cited by 2,366
- Polynomial.Xproof · cited by 1,639
- Polynomial.Cproof · cited by 1,598
- Finset.rangeproof · cited by 1,341
- Polynomial.natDegreestatement and proof · cited by 1,105
- Polynomial.coeffstatement and proof · cited by 1,045
Cited by1
Results whose statement or proof uses this declaration.
- WeierstrassCurve.Affine.irreducible_polynomialproof · cited by 0