(i) State Dirichlet's unit theorem.

(ii) Let $K$ be a number field. Show that if every conjugate of $\alpha \in \mathcal{O}_{K}$ has absolute value at most 1 then $\alpha$ is either zero or a root of unity.

(iii) Let $k=\mathbb{Q}(\sqrt{3})$ and $K=\mathbb{Q}(\zeta)$ where $\zeta=e^{i \pi / 6}=(i+\sqrt{3}) / 2$. Compute $N_{K / k}(1+\zeta)$. Show that

$$\mathcal{O}_{K}^{*}=\left\{(1+\zeta)^{m} u: 0 \leqslant m \leqslant 11, u \in \mathcal{O}_{k}^{*}\right\}$$

Hence or otherwise find fundamental units for $k$ and $K$.

[You may assume that the only roots of unity in $K$ are powers of $\zeta .$ ]