Theorems · Theorem · field theory
Polynomial.natDegree_comp
∀ {R : Type u} [inst : Semiring R] {p q : Polynomial R} [NoZeroDivisors R],
(p.comp q).natDegree = p.natDegree * q.natDegree- Defined in
- Mathlib.Algebra.Polynomial.Degree.Lemmas
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringNoZeroDivisors
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- Polynomialstatement and proof · cited by 5,681
- MulZeroClass.mul_zeroproof · cited by 2,091
- MulZeroClass.zero_mulproof · cited by 1,625
- Polynomial.Cproof · cited by 1,598
- Polynomial.natDegreestatement and proof · cited by 1,105
- Polynomial.coeffproof · cited by 1,045
- Polynomial.evalproof · cited by 796
- NoZeroDivisorsstatement and proof · cited by 545
- Polynomial.compstatement and proof · cited by 193
- Polynomial.natDegree_Cproof · cited by 59
Cited by13
Results whose statement or proof uses this declaration.
- minpoly.add_algebraMapproof · cited by 4
- Polynomial.leadingCoeff_compproof · cited by 4
- Polynomial.comp_eq_zero_iffproof · cited by 3
- Polynomial.natDegree_preHilbertPolyproof · cited by 3
- Polynomial.zero_lt_negOnePow_mul_eval_of_lt_roots_of_leadingCoeff_nonnegproof · cited by 2
- summable_pow_mul_geometric_of_norm_lt_oneproof · cited by 1
- cyclotomic_comp_X_add_one_isEisensteinAtproof · cited by 1
- cyclotomic_prime_pow_comp_X_add_one_isEisensteinAtproof · cited by 1
- Polynomial.irreducible_compproof · cited by 1
- Polynomial.isEquivalent_atBot_leadproof · cited by 1
- Polynomial.Gal.splits_in_splittingField_of_compproof · cited by 1
- Polynomial.natDegree_iterate_compproof · cited by 0