Euclidean domain

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) 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) (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.