(i) Show that the ring $k=\mathbf{F}_{2}[X] /\left(X^{2}+X+1\right)$ is a field. How many elements does it have?

(ii) Let $k$ be as in (i). By considering what happens to a chosen basis of the vector space $k^{2}$, or otherwise, find the order of the groups $G L_{2}(k)$ and $S L_{2}(k)$.

By considering the set of lines in $k^{2}$, or otherwise, show that $S L_{2}(k)$ is a subgroup of the symmetric group $S_{5}$, and identify this subgroup.