Euclidean domain

In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring (algebra) in which the Euclidean algorithm can be used. More precisely, a Euclidean domain is an integral domain ''D'' for which can be defined a function (mathematics) ''v'' mapping nonzero elements of ''D'' to non-negative integer and possessing the following properties: *For all nonzero ''a'' and ''b'' in ''D'', ''v''(''ab'') ≥ ''v''(''a''). *If ''a'' and ''b'' are in ''D'' and ''b'' is nonzero, then there are ''q'' and ''r'' in ''D'' such that ''a'' = ''qb'' + ''r'' and either ''r'' = 0 or ''v''(''r'') < ''v''(''b''). The function ''v'' is called a ''gauge'', ''valuation'' or ''norm''. Note that some authors define the function in an inequivalent way which nonetheless still gives the same class of rings. Examples of Euclidean domains include: *Z, the ring of integer. Define ''v''(''n'') = |''n''|, the absolute value of ''n''. *Z[''i''], the ring of Gaussian integer. Define ''v''(''z'') = |''z''|². *''K''[''X''], the polynomial ring over a field (mathematics) ''K''. For each nonzero polynomial ''f'', define ''v''(''f'') to be the degree of ''f''. *''K''''X'', the ring of formal power series over the field ''K''. For each nonzero power series ''f'', define ''v''(''f'') as the degree of the smallest power of ''X'' occurring in ''f''. *Any field. Define ''v''(''x'') = 1 for all nonzero ''x''. Every Euclidean domain is a principal ideal domain. In fact, if ''I'' is a nonzero ring ideal of a Euclidean domain ''D'' and a nonzero ''a'' in ''I'' is chosen to minimize ''v''(''a''), then ''I'' = ''aD''. The name comes from the fact that the extended Euclidean algorithm can be carried out in any Euclidean domain. abstract algebra

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Euclidean_domain