Theorems · Theorem · field theory
Irreducible.natDegree_pos
∀ {F : Type u_1} [inst : DivisionSemiring F] {f : Polynomial F}, Irreducible f → 0 < f.natDegreeAn irreducible polynomial over a field must have positive degree.
- Defined in
- Mathlib.Algebra.Polynomial.FieldDivision
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- DivisionSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Polynomialstatement and proof · cited by 5,681
- map_zeroproof · cited by 1,614
- Polynomial.Cproof · cited by 1,598
- Polynomial.natDegreestatement and proof · cited by 1,105
- Irreduciblestatement and proof · cited by 496
- DivisionSemiringstatement and proof · cited by 216
- Ne.isUnitproof · cited by 99
- Irreducible.not_isUnitproof · cited by 42
- Polynomial.isUnit_Cproof · cited by 19
- Polynomial.natDegree_eq_zeroproof · cited by 14
- not_irreducible_zeroproof · cited by 6
Cited by8
Results whose statement or proof uses this declaration.
- Irreducible.separableproof · cited by 6
- Irreducible.degree_posproof · cited by 2
- 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
- IsAlgClosure.of_splitsproof · cited by 1
- Polynomial.Separable.map_irreducible_of_isPurelyInseparableproof · cited by 0
- Irreducible.natDegree_dvd_iff_dvd_X_pow_card_pow_sub_Xproof · cited by 0
- IsAlgClosed.of_denseRangeproof · cited by 0