Euclidean domain

Revision as of 13:59, 22 August 2009 by Jam (talk | contribs) (Formal definition/examples)

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 $N(r)<N(b)$ or $r=0$.

Some common examples of Euclidean domains are:

See also

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