Let $K = F(a)$ be a finite extension of $F$. For $\alpha \in K$, let $L_{\alpha}$ be the map from $K$ to $K$ defined by $L_{\alpha}(x) = \alpha x$. Show that $L_{\alpha}$ is an $F$-linear transformation. Also show that $\det(xI - L_{\alpha})$ is the minimal polynomial $\min(F,a)$ of $a$. For which $\alpha \in K$ is $\det(xI-L_{\alpha})=\min(F,\alpha)$?
I showed that $L_{\alpha}$ is a $F$-linear transformation. How I show that $\det(xI - L_{\alpha}) = \min(F,a)$? I like any hint, no solutions. Thanks for the advance!
Hint: $P=det(xI-L_\alpha)(L_\alpha)=0$ Cayley-Hamilton, $P(L_\alpha)(1)=0$, deduce that $\alpha$ is a root of $P$, if $P=QR$ show that $\alpha$ is a root of $Q$ or $R$, deduce that $[K:F]$is the degree of $Q$ or $R$. Deduce that $Q=P$ or $R=P$ up to a scalar.