Theorems · Theorem · field theory

Polynomial.Monic.irreducible_iff_natDegree

∀ {R : Type u} [inst : CommSemiring R] [NoZeroDivisors R] {p : Polynomial R},
  p.Monic →
    (Irreducible p ↔ p ≠ 1 ∧ ∀ (f g : Polynomial R), f.Monic → g.Monic → f * g = p → f.natDegree = 0 ∨ g.natDegree = 0)

Defined in: Mathlib.Algebra.Polynomial.Monic
Cited by: 1 results in Mathlib
Foundations: Depth 109 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommSemiring NoZeroDivisors

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Cites8

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CommSemiringstatement and proof · cited by 10,911
Polynomialstatement and proof · cited by 5,681
Polynomial.natDegreestatement and proof · cited by 1,105
NoZeroDivisorsstatement and proof · cited by 545
Irreduciblestatement and proof · cited by 496
Polynomial.Monicstatement and proof · cited by 461
Polynomial.Monic.natDegree_eq_zeroproof · cited by 10
Polynomial.irreducible_of_monicproof · cited by 3

Cited by1

Results whose statement or proof uses this declaration.

Polynomial.Monic.irreducible_iff_natDegree'proof · cited by 2