Theorems · Theorem · field theory
Polynomial.natDegree_modByMonic_lt
∀ {R : Type u} [inst : Ring R] (p : Polynomial R) {q : Polynomial R}, q.Monic → q ≠ 1 → (p %ₘ q).natDegree < q.natDegree- Defined in
- Mathlib.Algebra.Polynomial.Div
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Ring
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.
- Ringstatement and proof · cited by 7,463
- Polynomialstatement and proof · cited by 5,681
- Nontrivialproof · cited by 2,416
- Polynomial.natDegreestatement and proof · cited by 1,105
- Polynomial.Monicstatement and proof · cited by 461
- Polynomial.modByMonicstatement and proof · cited by 82
- Polynomial.degree_modByMonic_ltproof · cited by 27
- Polynomial.natDegree_zeroproof · cited by 24
- Polynomial.Nontrivial.of_polynomial_neproof · cited by 10
- Polynomial.eq_one_of_monic_natDegree_zeroproof · cited by 6
- Polynomial.natDegree_lt_natDegreeproof · cited by 5
Cited by4
Results whose statement or proof uses this declaration.
- Submodule.span_range_natDegree_eq_adjoinproof · cited by 3
- exists_isIntegral_sub_of_isIntegralElem_of_mul_mem_rangeproof · cited by 1
- Polynomial.resultant_eq_prod_roots_subproof · cited by 1
- Polynomial.natDegree_mod_ltproof · cited by 0