# mathlib3documentation

field_theory.is_alg_closed.spectrum

# Spectrum mapping theorem #

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This file develops proves the spectral mapping theorem for polynomials over algebraically closed fields. In particular, if a is an element of an 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_is_alg_closed_of_finite_dimensional: 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_is_unit_aeval_prod {R : Type u} {A : Type v} [comm_ring R] [ring A] [ A] [is_domain R] {p : polynomial R} {a : A} (hp : p 0) (h : ¬is_unit ( (multiset.map (λ (x : R), p.roots).prod)) :
(k : R), k a = 0
theorem spectrum.subset_polynomial_aeval {𝕜 : Type u} {A : Type v} [field 𝕜] [ring A] [ A] (a : A) (p : polynomial 𝕜) :
(λ (k : 𝕜), p) '' 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] [ A] (a : A) (p : polynomial 𝕜) (hdeg : 0 < p.degree) :
( p) = (λ (k : 𝕜), p) '' 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] [ A] (a : A) (p : polynomial 𝕜) (hnon : (spectrum 𝕜 a).nonempty) :
( p) = (λ (k : 𝕜), p) '' 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] [ A] (a : A) (n : ) :
(λ (x : 𝕜), x ^ n) '' a (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] [ A] (a : A) {n : } (hn : 0 < n) :
(a ^ n) = (λ (x : 𝕜), x ^ n) '' 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] [ A] {a : A} (ha : (spectrum 𝕜 a).nonempty) (n : ) :
(a ^ n) = (λ (x : 𝕜), x ^ n) '' a

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

theorem spectrum.nonempty_of_is_alg_closed_of_finite_dimensional (𝕜 : Type u) {A : Type v} [field 𝕜] [ring A] [ A] [nontrivial A] [I : A] (a : A) :

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