Theorems · Theorem · field theory
minpoly.eq_of_irreducible
∀ {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 * Polynomial.C p.leadingCoeff⁻¹ = minpoly A x- Defined in
- Mathlib.FieldTheory.Minpoly.Field
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 122 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.
Cites25
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
- RingHomstatement · cited by 10,189
- Ringstatement and proof · cited by 7,463
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- AlgHomstatement · cited by 3,236
- Nontrivialstatement and proof · cited by 2,416
- MulZeroClass.zero_mulproof · cited by 1,625
- Polynomial.Cstatement and proof · cited by 1,598
- Polynomial.aevalstatement and proof · cited by 615
- Polynomial.leadingCoeffstatement and proof · cited by 498
Cited by8
Results whose statement or proof uses this declaration.
- exists_isTranscendenceBasis_and_isSeparable_of_linearIndepOn_powproof · cited by 2
- Irreducible.natDegree_le_twoproof · cited by 2
- minpoly.Irreducible.eq_minpolyproof · cited by 1
- RatFunc.finrank_eq_max_natDegreeproof · cited by 1
- isSolvable_gal_of_irreducibleproof · cited by 1
- Irreducible.dvd_iff_aeval_eq_zeroproof · cited by 1
- Polynomial.Gal.galAction_isPretransitiveproof · cited by 0
- Polynomial.Separable.map_irreducible_of_isPurelyInseparableproof · cited by 0