Theorems · Theorem · field theory
Polynomial.IsMonicOfDegree.eq_isMonicOfDegree_one_mul_isMonicOfDegree
∀ {F : Type u_1} [inst : Field F] [IsAlgClosed F] {f : Polynomial F} {n : ℕ},
f.IsMonicOfDegree (n + 1) → ∃ f₁ f₂, f₁.IsMonicOfDegree 1 ∧ f₂.IsMonicOfDegree n ∧ f = f₁ * f₂If f : F[X] is monic of degree ≥ 1 and F is an algebraically closed field,
then f = f₁ * f₂ with f₁ monic of degree 1 and f₂ monic of degree f.natDegree - 1.
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- Foundations
- Depth 127 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldIsAlgClosed
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Cites12
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
- add_commproof · cited by 1,535
- Irreducibleproof · cited by 496
- Polynomial.Monicproof · cited by 461
- IsAlgClosedstatement and proof · cited by 150
- Polynomial.natDegree_eq_of_degree_eq_someproof · cited by 42
- Polynomial.IsMonicOfDegreestatement and proof · cited by 39
- Polynomial.not_isUnit_of_natDegree_posproof · cited by 5
- Polynomial.exists_monic_irreducible_factorproof · cited by 3
- IsAlgClosed.degree_eq_one_of_irreducibleproof · cited by 2
- Polynomial.IsMonicOfDegree.of_mul_leftproof · cited by 2
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