Theorems · Theorem · field theory
minpoly.ne_zero
∀ {A : Type u_1} {B : Type u_2} [inst : CommRing A] [inst_1 : Ring B] [inst_2 : Algebra A B] {x : B} [Nontrivial A],
IsIntegral A x → minpoly A x ≠ 0A minimal polynomial is nonzero.
- Defined in
- Mathlib.FieldTheory.Minpoly.Basic
- Cited by
- 44 results in Mathlib
- Foundations
- Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- Polynomialstatement · cited by 5,681
- Nontrivialstatement and proof · cited by 2,416
- minpolystatement · cited by 439
- IsIntegralstatement and proof · cited by 427
- minpoly.monicproof · cited by 81
- Polynomial.Monic.ne_zeroproof · cited by 63
Cited by44
Results whose statement or proof uses this declaration.
- Algebra.isIntegral_traceproof · cited by 7
- minpoly.uniqueproof · cited by 7
- minpoly.add_algebraMapproof · cited by 4
- NumberField.Embeddings.finite_of_norm_leproof · cited by 4
- IntermediateField.finSepDegree_adjoin_simple_eq_finrank_iffproof · cited by 4
- minpoly.unique'proof · cited by 4
- Module.End.IsSemisimple.of_mem_adjoin_pairproof · cited by 3
- Field.nonempty_algHom_of_exists_rootproof · cited by 3
- minpoly.ne_zero_iffproof · cited by 3
- minpoly.ne_zero_of_finiteproof · cited by 3
- IsGalois.is_separable_splitting_fieldproof · cited by 2
- Module.End.hasEigenvalue_of_isRootproof · cited by 2