Suppose that $K$ is a number field of degree $n=r+2 s$, where $K$ has exactly real embeddings.

Show that the group of units in $\mathcal{O}_{K}$ is a finitely generated abelian group of rank at most $r+s-1$. Identify the torsion subgroup in terms of roots of unity.

[General results about discrete subgroups of a Euclidean real vector space may be used without proof, provided that they are stated clearly.]

Find all the roots of unity in $\mathbb{Q}(\sqrt{11})$.