Let $K$ be a number field of degree $n$, and let $\left\{\sigma_{i}: K \hookrightarrow \mathbb{C}\right\}$ be the set of complex embeddings of $K$. Show that if $\alpha \in \mathcal{O}_{K}$ satisfies $\left|\sigma_{i}(\alpha)\right|=1$ for every $i$, then $\alpha$ is a root of unity. Prove also that there exists $c>1$ such that if $\alpha \in \mathcal{O}_{K}$ and $\left|\sigma_{i}(\alpha)\right|<c$ for all $i$, then $\alpha$ is a root of unity.

State Dirichlet's Unit theorem.

Let $K \subset \mathbb{R}$ be a real quadratic field. Assuming Dirichlet's Unit theorem, show that the set of units of $K$ which are greater than 1 has a smallest element $\epsilon$, and that the group of units of $K$ is then $\left\{\pm \epsilon^{n} \mid n \in \mathbb{Z}\right\}$. Determine $\epsilon$ for $\mathbb{Q}(\sqrt{11})$, justifying your result. [If you use the continued fraction algorithm, you must prove it in full.]