Let $K=\mathbb{Q}(\sqrt{2})$.

(a) Write down the ring of integers $\mathcal{O}_{K}$.

(b) State Dirichlet's unit theorem, and use it to determine all elements of the group of units $\mathcal{O}_{K}^{\times}$.

(c) Let $P \subset \mathcal{O}_{K}$ denote the ideal generated by $3+\sqrt{2}$. Show that the group

$$G=\left\{\alpha \in \mathcal{O}_{K}^{\times} \mid \alpha \equiv 1 \bmod P\right\}$$

is cyclic, and find a generator.