# 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 (fun 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} [] [Ring A] [Algebra R A] [] {p : } {a : A} (h : ¬IsUnit (() (Multiset.map (fun (x : R) => Polynomial.X - Polynomial.C x) p.roots).prod)) :
kspectrum R a, = 0
theorem spectrum.subset_polynomial_aeval {𝕜 : Type u} {A : Type v} [] [Ring A] [Algebra 𝕜 A] (a : A) (p : ) :
(fun (x : 𝕜) => ) '' spectrum 𝕜 a spectrum 𝕜 (() 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} [] [Ring A] [Algebra 𝕜 A] [] (a : A) (p : ) (hdeg : 0 < p.degree) :
spectrum 𝕜 (() p) = (fun (x : 𝕜) => ) '' 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} [] [Ring A] [Algebra 𝕜 A] [] (a : A) (p : ) (hnon : (spectrum 𝕜 a).Nonempty) :
spectrum 𝕜 (() p) = (fun (k : 𝕜) => ) '' 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} [] [Ring A] [Algebra 𝕜 A] (a : A) (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} [] [Ring A] [Algebra 𝕜 A] [] (a : A) {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} [] [Ring A] [Algebra 𝕜 A] [] {a : A} (ha : (spectrum 𝕜 a).Nonempty) (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} [] [Ring A] [Algebra 𝕜 A] [] [] [I : ] (a : A) :
(spectrum 𝕜 a).Nonempty

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