Theorems · Theorem · field theory
minpoly.irreducible
∀ {A : Type u_1} {B : Type u_2} [inst : CommRing A] [inst_1 : Ring B] [inst_2 : Algebra A B] {x : B} [IsDomain A]
[IsDomain B], IsIntegral A x → Irreducible (minpoly A x)A minimal polynomial is irreducible.
- Defined in
- Mathlib.FieldTheory.Minpoly.Basic
- Cited by
- 26 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.
Cites22
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
- Ringstatement and proof · cited by 7,463
- Polynomialstatement and proof · cited by 5,681
- mul_commproof · cited by 2,262
- IsDomainstatement and proof · cited by 2,196
- IsUnitproof · cited by 1,602
- Polynomial.aevalproof · cited by 615
- Irreduciblestatement · cited by 496
- Polynomial.Monicproof · cited by 461
- minpolystatement and proof · cited by 439
Cited by26
Results whose statement or proof uses this declaration.
- isPurelyInseparable_iff_pow_memproof · cited by 10
- minpoly.isIntegrallyClosed_eq_field_fractionsproof · cited by 6
- IsAlgClosed.algebraMap_bijective_of_isIntegralproof · cited by 6
- minpoly.eq_iff_aeval_minpoly_eq_zeroproof · cited by 2
- IsSepClosed.algebraMap_surjectiveproof · cited by 2
- Irreducible.exists_dvd_monic_irreducible_of_isIntegralproof · cited by 2
- Algebra.trace_eq_zero_of_not_isSeparableproof · cited by 2
- exists_isTranscendenceBasis_and_isSeparable_of_linearIndepOn_powproof · cited by 2
- isConjRoot_of_aeval_eq_zeroproof · cited by 2
- gal_X_pow_sub_C_isSolvable_auxproof · cited by 1
- irreducible_X_pow_sub_C_of_root_adjoin_eq_topproof · cited by 1
- minpoly.map_eq_of_equiv_equivproof · cited by 1