Lemma 9.20.2. Let $L/K$ be a finite extension of fields. Let $\alpha \in L$ and let $P$ be the minimal polynomial of $\alpha$ over $K$. Then the characteristic polynomial of the $K$-linear map $\alpha : L \to L$ is equal to $P^ e$ with $e \deg (P) = [L : K]$.

Proof. Choose a basis $\beta _1, \ldots , \beta _ e$ of $L$ over $K(\alpha )$. Then $e$ satisfies $e \deg (P) = [L : K]$ by Lemmas 9.9.2 and 9.7.7. Then we see that $L = \bigoplus K(\alpha ) \beta _ i$ is a direct sum decomposition into $\alpha$-invariant subspaces hence the characteristic polynomial of $\alpha : L \to L$ is equal to the characteristic polynomial of $\alpha : K(\alpha ) \to K(\alpha )$ to the power $e$.

To finish the proof we may assume that $L = K(\alpha )$. In this case by Cayley-Hamilton we see that $\alpha$ is a root of the characteristic polynomial. And since the characteristic polynomial has the same degree as the minimal polynomial, we find that equality holds. $\square$

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