Roots of a minimal polynomial

Let $𝐴$ be a $𝕂$ -monoid over $𝕂$ and $𝑎 \in 𝐴$ be an algebraic element with minimal polynomial $𝑚 𝑎 (𝑥) \in 𝕂 [𝑥]$ . Then $𝑟 \in 𝕂$ is a root of $𝑚 𝑎 (𝑥)$ iff $𝑎 - 𝑟 1$ is not invertible in $𝐴$ .¹ #m/thm/falg