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Define a Euclidean Domain.

A Euclidean Domain, R\,, is an integral domain with the properties:

1. There is a function \varphi :R\backslash \{0\}\longrightarrow {\mathbb  {Z}}\geq 0 such that for all non-zero a,b\in R, \varphi (a)\leq \varphi (ab).

2. For all a,b\in R, with b\neq 0,\exists q,r\in R such that a=qb+r\,, where r=0\, or \varphi (r)<\varphi (b).

Main Page : Abstract Algebra : Euclidean Domains