(a) Let $L$ be a number field, $\mathcal{O}_{L}$ the ring of integers in $L, \mathcal{O}_{L}^{*}$ the units in $\mathcal{O}_{L}, r$ the number of real embeddings of $L$, and $s$ the number of pairs of complex embeddings of $L$.

Define a group homomorphism $\mathcal{O}_{L}^{*} \rightarrow \mathbb{R}^{r+s-1}$ with finite kernel, and prove that the image is a discrete subgroup of $\mathbb{R}^{r+s-1}$.

(b) Let $K=\mathbb{Q}(\sqrt{d})$ where $d>1$ is a square-free integer. What is the structure of the group of units of $K$ ? Show that if $d$ is divisible by a prime $p \equiv 3(\bmod 4)$ then every unit of $K$ has norm $+1$. Find an example of $K$ with a unit of norm $-1$.