Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 2 / 2G / Solution

A Euclidean domain is an integral domain $R$ with a function $\delta :R\setminus \{0\}\to {\mathbb{Z}}_{\ge 0}$ such that, for every $a\in R$ and nonzero $b\in R$, there are $q,r\in R$ satisfying $a=bq+r$, with either $r=0$ or $\delta (r)<\delta (b)$.

For the integers, take $\delta (b)=|b|$. Given $a$ and $b\ne 0$, put $m=|b|$ and choose the integer $q_0=\lfloor a/m\rfloor$. Then $a=m{q}_0+r$ with $0\le r<m$. Setting $q=q_0$ when $b>0$ and $q=-q_0$ when $b<0$ gives $a=bq+r$. A nonzero remainder has $\delta (r)=r<|b|=\delta (b)$, and the integers are an integral domain. Thus $\mathbb{Z}$ is a Euclidean domain with Euclidean function $|b|$.