2008-11.md

course	course_year	question_number	tags	title	year
Groups, Rings and Modules	IB	11	IB 2008 Groups, Rings and Modules	$3 . \mathrm{II} . 11 \mathrm{G}$	2008

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 $N(z)=z \bar{z}$.