$\frac{d}{d Y}: K[Y] \rightarrow K[Y], \quad \sum_{i \geq 0} a_{i} Y^{i} \mapsto \sum_{i \geq 0}(i+1) a_{i+1} Y^{i}$
$P_n =\[f \in K[Y] \mid \operatorname{deg}(f) \leq n]$ (this is a set)
$f=\left. \frac{d}{dY} \right|_{P_n} : P_n \rightarrow P_n$
Task:
Let $p$ be a prime number, let $K=\mathbb{F}_{p}$, the field with $p$ elements. Determine the minimal polynomial and the Jordan normal form of $f$.
Hint: The integers $k, r \in \mathbb{Z}_{\geq 0}$ defined by $n=kp+r$ and $r<p$ play a role!
Problem/Approach:
Unfortunately, I got pretty stuck here. The hint isn't that attractive looking. Thanks for any help!