Theorems · Theorem · field theory
minpoly.degree_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).degreeThe degree of a minimal polynomial is positive.
- Defined in
- Mathlib.FieldTheory.Minpoly.Basic
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Ringstatement and proof · cited by 7,463
- Nontrivialstatement and proof · cited by 2,416
- WithBotstatement · cited by 1,498
- Polynomial.degreestatement · cited by 643
- minpolystatement · cited by 439
- IsIntegralstatement and proof · cited by 427
- Polynomial.natDegree_pos_iff_degree_posproof · cited by 22
- minpoly.natDegree_posproof · cited by 8
Cited by6
Results whose statement or proof uses this declaration.
- minpoly.prime_of_isIntegrallyClosedproof · cited by 1
- IntermediateField.Lifts.exists_lift_of_splits'proof · cited by 1
- minpoly.degree_eq_one_iffproof · cited by 1
- IsPrimitiveRoot.minpoly_eq_powproof · cited by 1
- Polynomial.lift_of_splitsproof · cited by 0
- Algebra.adjoin.powerBasis'_minpoly_genproof · cited by 0