Let $K$ be a number field, and $\mathcal{O}_{K}$ its ring of integers. Write down a characterisation of the units in $\mathcal{O}_{K}$ in terms of the norm. Without assuming Dirichlet's units theorem, prove that for $K$ a quadratic field the quotient of the unit group by $\{\pm 1\}$ is cyclic (i.e. generated by one element). Find a generator in the cases $K=\mathbb{Q}(\sqrt{-3})$ and $K=\mathbb{Q}(\sqrt{11})$.

Determine all integer solutions of the equation $x^{2}-11 y^{2}=n$ for $n=-1,5,14$.