(i) Let $K$ be any field and let $\lambda \in K$. Let $J_{\lambda, n}$ be the $n \times n$ Jordan block

$$J_{\lambda, n}=\left(\begin{array}{ccccc} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \ddots & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ \vdots & & & \ddots & 1 \\ 0 & \cdots & \cdots & 0 & \lambda \end{array}\right)$$

Compute $J_{\lambda, n}^{r}$ for each $r \geqslant 0$.

(ii) Let $G$ be a cyclic group of order $N$, and let $K$ be an algebraically closed field of characteristic $p \geqslant 0$. Determine all the representations of $G$ on vector spaces over $K$, up to equivalence. Which are irreducible? Which do not split as a direct sum $W \oplus W^{\prime}$, with $W \neq 0$ and $W^{\prime} \neq 0 ?$