Spectrum mapping theorem

This file develops proves the spectral mapping theorem for polynomials over algebraically closed fields. In particular, if a is an element of a 𝕜-algebra A where 𝕜 is a field, and p : 𝕜[X] is a polynomial, then the spectrum of Polynomial.aeval a p contains the image of the spectrum of a under (λ k, Polynomial.eval k p). When 𝕜 is algebraically closed, these are in fact equal (assuming either that the spectrum of a is nonempty or the polynomial has positive degree), which is the spectral mapping theorem.

In addition, this file contains the fact that every element of a finite dimensional nontrivial algebra over an algebraically closed field has nonempty spectrum. In particular, this is used in Module.End.exists_eigenvalue to show that every linear map from a vector space to itself has an eigenvalue.

Main statements

• spectrum.subset_polynomial_aeval, spectrum.map_polynomial_aeval_of_degree_pos, spectrum.map_polynomial_aeval_of_nonempty: variations on the spectral mapping theorem.
• spectrum.nonempty_of_isAlgClosed_of_finiteDimensional: the spectrum is nonempty for any element of a nontrivial finite dimensional algebra over an algebraically closed field.

Notations

• σ a : spectrum R a of a : A

theorem spectrum.exists_mem_of_not_isUnit_aeval_prod {R : Type u} {A : Type v} [CommRing R] [Ring A] [Algebra R A] [IsDomain R] {p : Polynomial R} {a : A} (h : ¬IsUnit ((Polynomial.aeval a) (Multiset.prod (Multiset.map (fun (x : R) => Polynomial.X - Polynomial.C x) (Polynomial.roots p))))) : ∃ k ∈ spectrum R a, Polynomial.eval k p = 0

theorem spectrum.subset_polynomial_aeval {𝕜 : Type u} {A : Type v} [Field 𝕜] [Ring A] [Algebra 𝕜 A] (a : A) (p : Polynomial 𝕜) : (fun (x : 𝕜) => Polynomial.eval x p) '' spectrum 𝕜 a ⊆ spectrum 𝕜 ((Polynomial.aeval a) p)

Half of the spectral mapping theorem for polynomials. We prove it separately because it holds over any field, whereas spectrum.map_polynomial_aeval_of_degree_pos and spectrum.map_polynomial_aeval_of_nonempty need the field to be algebraically closed.

theorem spectrum.map_polynomial_aeval_of_degree_pos {𝕜 : Type u} {A : Type v} [Field 𝕜] [Ring A] [Algebra 𝕜 A] [IsAlgClosed 𝕜] (a : A) (p : Polynomial 𝕜) (hdeg : 0 < Polynomial.degree p) : spectrum 𝕜 ((Polynomial.aeval a) p) = (fun (x : 𝕜) => Polynomial.eval x p) '' spectrum 𝕜 a

The spectral mapping theorem for polynomials. Note: the assumption degree p > 0 is necessary in case σ a = ∅, for then the left-hand side is ∅ and the right-hand side, assuming [Nontrivial A], is {k} where p = Polynomial.C k.

theorem spectrum.map_polynomial_aeval_of_nonempty {𝕜 : Type u} {A : Type v} [Field 𝕜] [Ring A] [Algebra 𝕜 A] [IsAlgClosed 𝕜] (a : A) (p : Polynomial 𝕜) (hnon : Set.Nonempty (spectrum 𝕜 a)) : spectrum 𝕜 ((Polynomial.aeval a) p) = (fun (k : 𝕜) => Polynomial.eval k p) '' spectrum 𝕜 a

In this version of the spectral mapping theorem, we assume the spectrum is nonempty instead of assuming the degree of the polynomial is positive.

theorem spectrum.pow_image_subset {𝕜 : Type u} {A : Type v} [Field 𝕜] [Ring A] [Algebra 𝕜 A] (a : A) (n : ℕ) : (fun (x : 𝕜) => x ^ n) '' spectrum 𝕜 a ⊆ spectrum 𝕜 (a ^ n)

A specialization of spectrum.subset_polynomial_aeval to monic monomials for convenience.

theorem spectrum.map_pow_of_pos {𝕜 : Type u} {A : Type v} [Field 𝕜] [Ring A] [Algebra 𝕜 A] [IsAlgClosed 𝕜] (a : A) {n : ℕ} (hn : 0 < n) : spectrum 𝕜 (a ^ n) = (fun (x : 𝕜) => x ^ n) '' spectrum 𝕜 a

A specialization of spectrum.map_polynomial_aeval_of_nonempty to monic monomials for convenience.

theorem spectrum.map_pow_of_nonempty {𝕜 : Type u} {A : Type v} [Field 𝕜] [Ring A] [Algebra 𝕜 A] [IsAlgClosed 𝕜] {a : A} (ha : Set.Nonempty (spectrum 𝕜 a)) (n : ℕ) : spectrum 𝕜 (a ^ n) = (fun (x : 𝕜) => x ^ n) '' spectrum 𝕜 a

A specialization of spectrum.map_polynomial_aeval_of_nonempty to monic monomials for convenience.

theorem spectrum.nonempty_of_isAlgClosed_of_finiteDimensional (𝕜 : Type u) {A : Type v} [Field 𝕜] [Ring A] [Algebra 𝕜 A] [IsAlgClosed 𝕜] [Nontrivial A] [I : FiniteDimensional 𝕜 A] (a : A) : Set.Nonempty (spectrum 𝕜 a)

Every element a in a nontrivial finite-dimensional algebra A over an algebraically closed field 𝕜 has non-empty spectrum.