Euclidean domain

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 $\delta(r)<\delta(b)$.

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).