Theorems · Theorem · commutative algebra
Polynomial.exists_monic_irreducible_factor
∀ {F : Type u_1} [inst : Field F] (f : Polynomial F), ¬IsUnit f → ∃ g, g.Monic ∧ Irreducible g ∧ g ∣ fA polynomial over a field which is not a unit must have a monic irreducible factor.
See also WfDvdMonoid.exists_irreducible_factor.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 126 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Field
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- Polynomial.Xproof · cited by 1,639
- IsUnitstatement and proof · cited by 1,602
- Polynomial.Cproof · cited by 1,598
- Polynomial.leadingCoeffproof · cited by 498
- Irreduciblestatement and proof · cited by 496
- Polynomial.Monicstatement · cited by 461
- Associatedproof · cited by 296
- Ne.isUnitproof · cited by 99
- dvd_zeroproof · cited by 63
Cited by3
Results whose statement or proof uses this declaration.
- Polynomial.Monic.eq_X_pow_char_pow_sub_C_pow_of_natSepDegree_eq_oneproof · cited by 1
- Polynomial.IsMonicOfDegree.eq_isMonicOfDegree_one_or_two_mulproof · cited by 1
- Polynomial.IsMonicOfDegree.eq_isMonicOfDegree_one_mul_isMonicOfDegreeproof · cited by 0