Theorems · Theorem · field theory
minpoly.eq_of_irreducible_of_monic
∀ {A : Type u_1} {B : Type u_2} [inst : Field A] [inst_1 : Ring B] [inst_2 : Algebra A B] {x : B} [Nontrivial B]
{p : Polynomial A}, Irreducible p → (Polynomial.aeval x) p = 0 → p.Monic → p = minpoly A x- Defined in
- Mathlib.FieldTheory.Minpoly.Field
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 121 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldRingAlgebraNontrivial
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Algebrastatement and proof · cited by 11,388
- Ringstatement and proof · cited by 7,463
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- mul_oneproof · cited by 3,885
- AlgHomstatement · cited by 3,236
- Nontrivialstatement and proof · cited by 2,416
- Polynomial.aevalstatement and proof · cited by 615
- Irreduciblestatement and proof · cited by 496
- Polynomial.Monicstatement and proof · cited by 461
- minpolystatement and proof · cited by 439
Cited by9
Results whose statement or proof uses this declaration.
- minpoly.eq_of_irreducibleproof · cited by 8
- minpoly.isIntegrallyClosed_eq_field_fractionsproof · cited by 6
- IsPrimitiveRoot.minpoly_eq_cyclotomic_of_irreducibleproof · cited by 4
- isConjRoot_of_aeval_eq_zeroproof · cited by 2
- minpoly.eq_iff_aeval_eq_zeroproof · cited by 1
- Polynomial.irreducible_compproof · cited by 1
- minpoly.map_eq_of_equiv_equivproof · cited by 1
- FixedPoints.minpoly_eq_minpolyproof · cited by 0
- IsAlgClosed.of_denseRangeproof · cited by 0