Euclidean domainEuclideanRing2002-05-27 22:59:182007-12-28 16:58:36Definition\usepackage{amssymb}
\newcommand{\Z}{\mathbb{Z}}
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A \emph{Euclidean domain} is an integral domain
on which a Euclidean valuation can be defined.
Every Euclidean domain is a principal ideal domain,
and therefore also a unique factorization domain.
Any two elements of a Euclidean domain have a greatest common divisor,
which can be computed using the Euclidean algorithm.
An example of a Euclidean domain is the ring $\Z$.
Another example is the polynomial ring $F[x]$, where $F$ is any field.
Every field is also a Euclidean domain.