(a) Compute the class group of $K=\mathbb{Q}(\sqrt{30})$. Find also the fundamental unit of $K$, stating clearly any general results you use.

[The Minkowski bound for a real quadratic field is $\left|d_{K}\right|^{1 / 2} / 2 .$ ]

(b) Let $K=\mathbb{Q}(\sqrt{d})$ be real quadratic, with embeddings $\sigma_{1}, \sigma_{2} \hookrightarrow \mathbb{R}$. An element $\alpha \in K$ is totally positive if $\sigma_{1}(\alpha)>0$ and $\sigma_{2}(\alpha)>0$. Show that the totally positive elements of $K$ form a subgroup of the multiplicative group $K^{*}$ of index 4 .

Let $I, J \subset \mathcal{O}_{K}$ be non-zero ideals. We say that $I$ is narrowly equivalent to $J$ if there exists a totally positive element $\alpha$ of $K$ such that $I=\alpha J$. Show that this is an equivalence relation, and that the equivalence classes form a group under multiplication. Show also that the order of this group equals

$$\begin{cases}\text { the class number } h_{K} \text { of } K & \text { if the fundamental unit of } K \text { has norm }-1 \\ 2 h_{K} & \text { otherwise. }\end{cases}$$