Spectrum of an algebraic element

Spectral mapping theorem

Let 𝐴 be a K-monoid over an algebraically closed field 𝕂. Let 𝑎 ∈𝐴 and 𝑝(𝑥) ∈𝕂[𝑥]. Then the spectra satisfy¹ falg

Spec⁡(𝑝(𝑎))=𝑝(Spec⁡(𝑎))={𝑝(𝜆),𝜆∈Spec⁡(𝑎)}

Proof

proof

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Footnotes

1. Stated with partial proof in 2008. Advanced Linear Algebra, §18, p. 461 ↩