 Solution

A Euclidean domain is an integral domain equipped with a function $\delta :R\setminus \{0\}\to \mathbb{N}$ such that for every $a\in R$ and nonzero $b\in R$ there are $q,r\in R$ with $a=qb+r$ and either $r=0$ or $\delta (r)<\delta (b)$. One may additionally require $\delta (a)\le \delta (ab)$ for nonzero $a,b$; degree on a polynomial ring also satisfies this convention.