# Euclidean domain

A 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 $\mathbb{Z}$. Another example is the polynomial ring $F[x]$, where $F$ is any field. Every field is also a Euclidean domain.

 Title Euclidean domain Canonical name EuclideanDomain Date of creation 2013-03-22 12:40:42 Last modified on 2013-03-22 12:40:42 Owner yark (2760) Last modified by yark (2760) Numerical id 13 Author yark (2760) Entry type Definition Classification msc 13F07 Synonym Euclidean ring Related topic PID Related topic UFD Related topic EuclidsAlgorithm Related topic Ring Related topic IntegralDomain Related topic EuclideanValuation Related topic WhyEuclideanDomains