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**Bruno J.** Let $\displaystyle K$ be a field, $\displaystyle [K:\mathbb{Q}]=n$, and take $\displaystyle \alpha \in K$ such that $\displaystyle \alpha$ has degree $\displaystyle d$ over $\displaystyle \mathbb{Q}$. Let $\displaystyle f(x)$ be the monic irreducible polynomial over $\displaystyle \mathbb{Q}$ having $\displaystyle \alpha$ as a root.

Let $\displaystyle T_\alpha : K \rightarrow K$ be given by $\displaystyle k \mapsto \alpha k$. Show that the characteristic polynomial of $\displaystyle T_\alpha$ is $\displaystyle f(x)^{n/d}$.