Theorems · Theorem · field theory
Polynomial.IsMonicOfDegree.eq_isMonicOfDegree_one_or_two_mul
∀ {f : Polynomial ℝ} {n : ℕ},
f.IsMonicOfDegree (n + 1) → ∃ f₁ f₂, (f₁.IsMonicOfDegree 1 ∨ f₁.IsMonicOfDegree 2) ∧ f = f₁ * f₂If f : ℝ[X] is monic of positive degree, then f = f₁ * f₂ with f₁ monic
of degree 1 or 2.
This relies on the fact that irreducible polynomials over ℝ have degree at most 2.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 297 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- Polynomialstatement and proof · cited by 5,681
- le_antisymmproof · cited by 2,068
- Polynomial.natDegreeproof · cited by 1,105
- Irreducibleproof · cited by 496
- Polynomial.Monicproof · cited by 461
- Polynomial.IsMonicOfDegreestatement and proof · cited by 39
- Irreducible.natDegree_posproof · cited by 8
- Polynomial.not_isUnit_of_natDegree_posproof · cited by 5
- Polynomial.exists_monic_irreducible_factorproof · cited by 3
- Irreducible.natDegree_le_twoproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- Polynomial.IsMonicOfDegree.eq_isMonicOfDegree_two_mul_isMonicOfDegreeproof · cited by 0