Euclidean domain - Imported Knowlpedia Content

A Euclidean domain is an integral domain $R$ equipped with a function $\delta:R\setminus\{0\}\to \mathbb{N}$ such that for all $a\in R$ and $b\in R\setminus\{0\}$ , there exist $q,r\in R$ with $a=bq+r$ and either $r=0$ or $\delta(r)<\delta(b)$ .

Remarks

This “division algorithm” implies the Euclidean algorithm and ensures existence of gcds. In particular, every Euclidean domain is a PID (see Euclidean implies PID).

Examples

$\mathbb{Z}$ with $\delta(n)=|n|$ is Euclidean.
If $k$ is a field, then $k[x]$ is Euclidean with $\delta(f)=\deg(f)$ for $f\neq 0$ .
$k[x,y]$ is not Euclidean (it is not even a PID).