Theorems · Theorem · commutative algebra
Polynomial.modByMonic_eq_zero_iff_quotient_eq_zero
∀ {R : Type u_1} [inst : CommRing R] (p q : Polynomial R),
q.Monic → (p %ₘ q = 0 ↔ (Ideal.Quotient.mk (Ideal.span {q})) p = 0)- Defined in
- Mathlib.RingTheory.Polynomial.Quotient
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRing
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Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- RingHomstatement · cited by 10,189
- Polynomialstatement and proof · cited by 5,681
- Idealstatement · cited by 4,748
- HasQuotient.Quotientstatement · cited by 2,301
- Ideal.spanstatement and proof · cited by 948
- Ideal.Quotient.mkstatement and proof · cited by 610
- Polynomial.Monicstatement and proof · cited by 461
- Polynomial.modByMonicstatement · cited by 82
- Polynomial.modByMonic_eq_zero_iff_dvdproof · cited by 15
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