What is a Euclidean domain? Show that a Euclidean domain is a principal ideal domain.

Show that $\mathbb{Z}[\sqrt{-7}]$ is not a Euclidean domain (for any choice of norm), but that the ring

$$\mathbb{Z}\left[\frac{1+\sqrt{-7}}{2}\right]$$

is Euclidean for the norm function $N(z)=z \bar{z}$.