Euclidean domain

A Euclidean domain is an integral domainon 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