Theorems · Theorem · field theory
isConjRoot_iff_mem_minpoly_aroots
∀ {K : Type u_2} {S : Type u_4} [inst : CommRing S] [inst_1 : Field K] [inst_2 : Algebra K S] [inst_3 : IsDomain S]
{x y : S}, IsIntegral K x → (IsConjRoot K x y ↔ y ∈ (minpoly K x).aroots S)y is a conjugate root of x over K if and only if y is a root of the minimal polynomial of
x. This is variant of isConjRoot_iff_aeval_eq_zero.
- Defined in
- Mathlib.FieldTheory.Minpoly.IsConjRoot
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 135 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- Fieldstatement and proof · cited by 7,404
- Multisetstatement · cited by 2,627
- IsDomainstatement and proof · cited by 2,196
- Polynomial.aevalproof · cited by 615
- minpolystatement and proof · cited by 439
- IsIntegralstatement and proof · cited by 427
- Polynomial.arootsstatement · cited by 89
- minpoly.ne_zeroproof · cited by 44
- IsConjRootstatement and proof · cited by 43
Cited by1
Results whose statement or proof uses this declaration.
- isConjRoot_iff_mem_minpoly_rootSetproof · cited by 2