Deduce that there exist $r$ vector subspaces of $E$, denoted $E_1, \ldots, E_r$, all stable under $f$ such that:
\begin{itemize}
\item $E = E_1 \oplus \cdots \oplus E_r$;
\item for all $1 \leqslant i \leqslant r$, the endomorphism $\psi_i$ induced by $f$ on the vector subspace $E_i$ is cyclic;
\item if we denote by $P_i$ the minimal polynomial of $\psi_i$, then $P_{i+1}$ divides $P_i$ for all integer $i$ such that $1 \leqslant i \leqslant r-1$.
\end{itemize}