I am asked to show that if $L=K(\beta)$, prove that the minimal polynomial of $\beta$ is the characteristic polynomial of $T(\beta)$, with leading coefficient 1 (monic).
This is part (iii) of a question, part (i) and (ii) I have done, and I list it below in case it is needed for part (iii)
(i) Assume $[L:K]=n$. For any $\beta\in L$, show that the map $T_\beta(k)=\beta k$ is a linear transformation (scalars in K) on the vector space L. Hence, show that $L$ is isomorphic to a subring of $M_n(K)$.
(ii) Show that $Tr(\beta)=Trace(T(\beta))$ and $N(\beta)=det(T(\beta))$ are additive and multiplicative (respectively) group homomorphisms.