# 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 |