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Euclidean domain

From AoPSWiki

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:

The ring of integers $\mathbb Z$ with norm given by $N(a) = |a|$ .
The ring of Gaussian integers $\mathbb Z[i]$ with norm given by $N(a+bi) = a^2+b^2$ .
The ring of polynomials $F[x]$ over any field $F$ with norm given by $N(p) = \deg p$ .

It can be easily shown through infinite descent that any Euclidian domain is also a principal ideal domain. Indeed, let $I$ be any ideal of a Euclidean domain $R$ and take some $a\in I$ with minimal norm. We claim that $I=(a)$ . Clearly $(a)\subseteq I$ , because $I$ is an ideal. Now assume $(a)\ne I$ and consider any $b\in I\setminus (a)$ . Applying the division algorithm we get that there are $q,r\in R$ such that $b=aq+r$ with $N(r)<N(a)$ (we cannot have $r=0$ as $b\not\in (a)$ ). But now as $I$ is an ideal, and $a,b\in I$ , we must have $r=b-aq\in I$ , contradicting the minimality of $a$ . Hence $I=(a)$ and $R$ is indeed a principle ideal domain.

See also

Wikipedia: Euclidean Domain

Mathworld: Euclidean Ring

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