(i) Suppose that $d>1$ is a square-free integer. Describe, with justification, the ring of integers in the field $K=\mathbb{Q}(\sqrt{d})$.

(ii) Show that $\mathbb{Q}\left(2^{1 / 3}\right)=\mathbb{Q}\left(4^{1 / 3}\right)$ and that $\mathbb{Z}\left[4^{1 / 3}\right]$ is not the ring of integers in this field.