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Let $K$ be a field extension of $F$ with degree $n$. Let the map $T_{\alpha}$ for $\alpha \in K$ be $T_{\alpha}:K \rightarrow K$, $T_{\alpha}:k \mapsto \alpha k$. Show that there is an injective ring homomorphism $K \rightarrow M_{n}(F)$ and that $\alpha$ is a root of the characteristic polynomial of $M_{\alpha}$ for $\alpha$ given.


I first showed that $T_{\alpha}$ is a linear map and that the injective homomorphism of rings holds. However, how could one show the other statement? I tried to show at the beginning that $\alpha$ was an eigenvalue of the map (but clearly this isn't the way since the scalars in this case would be elements in $F$). What would be a hint to proceed?

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Hint: Use Cayley-Hamilton together with the fact that $\alpha\mapsto T_\alpha$ is a homomorphism of $F$-algebras, so it preserves evaluation of polynomials with coefficients in $F$.

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