Theorems · Theorem · commutative algebra
AdjoinRoot.mk_ne_zero_of_natDegree_lt
∀ {R : Type u_1} [inst : CommRing R] {f : Polynomial R},
f.Monic → ∀ {g : Polynomial R}, g ≠ 0 → g.natDegree < f.natDegree → (AdjoinRoot.mk f) g ≠ 0- Defined in
- Mathlib.RingTheory.AdjoinRoot
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- RingHomstatement · cited by 10,189
- Polynomialstatement and proof · cited by 5,681
- Polynomial.natDegreestatement and proof · cited by 1,105
- Iff.notproof · cited by 489
- Polynomial.Monicstatement and proof · cited by 461
- AdjoinRootstatement · cited by 177
- AdjoinRoot.mkstatement · cited by 50
- AdjoinRoot.mk_eq_zeroproof · cited by 4
- Polynomial.Monic.not_dvd_of_natDegree_ltproof · cited by 3
Cited by4
Results whose statement or proof uses this declaration.
- root_X_pow_sub_C_ne_zeroproof · cited by 2
- WeierstrassCurve.Affine.CoordinateRing.XClass_ne_zeroproof · cited by 2
- WeierstrassCurve.Affine.CoordinateRing.YClass_ne_zeroproof · cited by 1
- root_X_pow_sub_C_ne_zero'proof · cited by 1