Theorems · Theorem · commutative algebra
Polynomial.isPrimitive_iff_contentIdeal_eq_top
∀ {R : Type u_3} [inst : CommSemiring R] [IsBezout R] (p : Polynomial R), p.IsPrimitive ↔ p.contentIdeal = ⊤The polynomial p is primitive if and only if the coefficients of p generate the whole ring.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 107 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringIsBezout
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- Top.topstatement · cited by 9,680
- Polynomialstatement and proof · cited by 5,681
- Idealstatement · cited by 4,748
- Polynomial.IsPrimitivestatement · cited by 33
- Polynomial.contentIdealstatement and proof · cited by 24
- IsBezoutstatement and proof · cited by 23
- IsBezout.isPrincipal_of_FGproof · cited by 2
- Submodule.IsPrincipal.isPrimitive_iff_contentIdeal_eq_topproof · cited by 1
- Polynomial.contentIdeal_FGproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- Polynomial.isPrimitive_iff_forall_gaussNorm_eq_oneproof · cited by 0