# Difference between revisions of "Euclidean domain"

A Euclidean domain (or Euclidean ring) is a type of ring in which the Euclidean algorithm can be used.

Formally we say that a ring $R$ is a Euclidean domain if:

• It is an integral domain.
• There a function $N:R\setminus\{0\}\to \mathbb Z_{\ge0}$ called a Norm such that for all nonzero $a,b\in R$ there are $q,r\in R$ such that $a = bq+r$ and either $N(r) or $r=0$.

Some common examples of Euclidean domains are:

• The ring of integers $\mathbb Z$ with norm given by $N(a) = |a|$.
• The ring of Gaussian integers $\mathbb Z[i]$ with norm given by $N(a+bi) = a^2+b^2$.
• The ring of polynomials $F[x]$ over any field $F$ with norm given by $N(p) = \deg p$.