(a) Factorise the ideals [2], [3] and [5] in the ring of integers $\mathcal{O}_{K}$ of the field $K=\mathbb{Q}(\sqrt{30})$. Using Minkowski's bound

$$\frac{n !}{n^{n}}\left(\frac{4}{\pi}\right)^{s} \sqrt{\left|d_{K}\right|},$$

determine the ideal class group of $K$.

[Hint: it might be helpful to notice that $\frac{3}{2}=N_{K / \mathbb{Q}}(\alpha)$ for some $\left.\alpha \in K .\right]$

(b) Find the fundamental unit of $K$ and determine all solutions of the equations $x^{2}-30 y^{2}=\pm 5$ in integers $x, y \in \mathbb{Z}$. Prove that there are in fact no solutions of $x^{2}-30 y^{2}=5$ in integers $x, y \in \mathbb{Z}$.