# Difference between revisions of "Euclidean domain"

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− | A '''Euclidean domain''' (or '''Euclidean ring''') is a type of [[ring]] in which the [[Euclidean algorithm]] can be used. | + | 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 <math>R</math> is a Euclidean domain if: | ||

+ | * It is an [[integral domain]]. | ||

+ | * There a function <math>N:R\setminus\{0\}\to \mathbb Z_{\ge0}</math> called a '''Norm''' such that for all nonzero <math>a,b\in R</math> there are <math>q,r\in R</math> such that <math>a = bq+r</math> and either <math>N(r)<N(b)</math> or <math>r=0</math>. | ||

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+ | Some common examples of Euclidean domains are: | ||

+ | * The ring of [[integers]] <math>\mathbb Z</math> with norm given by <math>N(a) = |a|</math>. | ||

+ | * The ring of [[Gaussian integers]] <math>\mathbb Z[i]</math> with norm given by <math>N(a+bi) = a^2+b^2</math>. | ||

+ | * The [[polynomial ring|ring of polynomials]] <math>F[x]</math> over any [[field]] <math>F</math> with norm given by <math>N(p) = \deg p</math>. | ||

==See also== | ==See also== |

## Revision as of 13:59, 22 August 2009

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 is a Euclidean domain if:

- It is an integral domain.
- There a function called a
**Norm**such that for all nonzero there are such that and either or .

Some common examples of Euclidean domains are:

- The ring of integers with norm given by .
- The ring of Gaussian integers with norm given by .
- The ring of polynomials over any field with norm given by .

## See also

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