[[Spectrum of an algebraic element]]
# Spectral mapping theorem

Let $A$ be a [[K-monoid]] over an [[algebraically closed field]] $\mathbb{K}$.
Let $a \in A$ and $p(x) \in \mathbb{K}[x]$.
Then the [[Spectrum of an algebraic element|spectra]] satisfy[^2008] #m/thm/falg 
$$
\begin{align*}
\Spec(p(a)) = p(\Spec(a)) = \{ p(\lambda), \lambda \in \Spec(a) \}
\end{align*}
$$

  [^2008]: Stated with partial proof in 2008\. [[Sources/@romanAdvancedLinearAlgebra2008|Advanced Linear Algebra]], §18, p. 461

> [!missing]- Proof
> #missing/proof

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