Remark 1

Clearly the minimal polynomial of the restriction of $u$ to $M_p$ is the greatest power of $p$ which divides the minimal polynomial of $u$. Moreover we have $s_p(u) \cdot x = x$ for $x \in M_p$, from which it follows immediately that, if $M_p \neq 0$, then $s_p$ is coprime to $p$.

Similarly, by Th. 2 of VII, p. 19, the module $E_u$ is isomorphic to a direct sum of cyclic modules $F_j = K[X]/a_j \ (1 \leq j \leq r)$, where the ideals $a_j$ are distinct from $K[X]$ and $a_j \subset a_{j+1}$; and the $a_j$ are determined by these conditions. Moreover, since $E$, is a torsion module, we have $a_1 \neq (0)$; since $E$ has dimension $n$, we have $r \leq n$. Put $a_j = (h_j) \ (1 \leq j \leq r)$, with $h_j$ a monic polynomial, and consider the sequence $(q_i) \ (1 \leq i \leq n)$ of polynomials defined by:

It is clear that the polynomials $q_i$ determine the polynomials $h_j$ and conversely, and that $E$, is isomorphic to the direct sum of the $n$ modules $K[X]/(q_i)$, the first $n - r$ of which are $0$.

In other words: