 Solution

A Euclidean domain is an integral domain $R$ admitting a Euclidean function $\delta :R\setminus \{0\}\to {\mathbb{Z}}_{\ge 0}$ such that division with a smaller remainder is possible: for every $a\in R$ and nonzero $b\in R$, there are $q,r\in R$ withThus the Euclidean function measures progress in repeated division. Definitions which additionally normalize it to be nondecreasing under multiplication give the same class of Euclidean domains; the division property is the one needed below.