Theorems · Theorem · field theory
minpoly.natDegree_pos
∀ {A : Type u_1} {B : Type u_2} [inst : CommRing A] [inst_1 : Ring B] [inst_2 : Algebra A B] {x : B} [Nontrivial B],
IsIntegral A x → 0 < (minpoly A x).natDegreeThe degree of a minimal polynomial, as a natural number, is positive.
- Defined in
- Mathlib.FieldTheory.Minpoly.Basic
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 112 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomproof · cited by 10,189
- Ringstatement and proof · cited by 7,463
- Polynomialproof · cited by 5,681
- Nontrivialstatement and proof · cited by 2,416
- Polynomial.Cproof · cited by 1,598
- Polynomial.natDegreestatement and proof · cited by 1,105
- Polynomial.coeffproof · cited by 1,045
- map_oneproof · cited by 861
- Polynomial.aevalproof · cited by 615
Cited by8
Results whose statement or proof uses this declaration.
- minpoly.degree_posproof · cited by 6
- minpoly.natSepDegree_eq_one_iff_pow_memproof · cited by 4
- IsIntegral.coeffproof · cited by 4
- spectralNorm_zero_ltproof · cited by 3
- Algebra.trace_eq_zero_of_not_isSeparableproof · cited by 2
- minpoly.two_le_natDegree_iffproof · cited by 2
- spectralNorm.spectralNorm_eq_norm_coeff_zero_rpowproof · cited by 0
- Valuation.pow_coeff_zero_ne_zero_of_unitproof · cited by 0