Theorems · Theorem · field theory
spectrum.exists_mem_of_not_isUnit_aeval_prod
∀ {R : Type u} {A : Type v} [inst : CommRing R] [inst_1 : Ring A] [inst_2 : Algebra R A] [inst_3 : IsDomain R]
{p : Polynomial R} {a : A},
¬IsUnit ((Polynomial.aeval a) (Multiset.map (fun x => Polynomial.X - Polynomial.C x) p.roots).prod) →
∃ k ∈ spectrum R a, Polynomial.eval k p = 0- Defined in
- Mathlib.FieldTheory.IsAlgClosed.Spectrum
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 133 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites29
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
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Ringstatement and proof · cited by 7,463
- Polynomialstatement and proof · cited by 5,681
- Algebra.algebraMapproof · cited by 4,706
- AlgHomstatement · cited by 3,236
- IsDomainstatement and proof · cited by 2,196
- Polynomial.Xstatement and proof · cited by 1,639
- IsUnitstatement and proof · cited by 1,602
Cited by2
Results whose statement or proof uses this declaration.
- spectrum.map_polynomial_aeval_of_degree_posproof · cited by 2
- spectrum.nonempty_of_isAlgClosed_of_finiteDimensionalproof · cited by 2