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More precisely, a Euclidean domain is an Integral Domain ''D'' on which one can define a Function ''v'' mapping nonzero elements of ''D'' to non-negative Integers that satisfies the following division-with-remainder property:
The function ''v'' is called a ''valuation'' or ''norm'' or ''gauge'' and the key point here is that the remainder ''r'' has ''v''-size smaller than the ''v''-size of the divisor ''b''. Nearly all algebra textbooks which discuss Euclidean domains include the following extra property in the definition: for all nonzero ''a'' and ''b'' in ''D'', ''v''(''ab'') ≥ ''v''(''a''). This property does not have to be assumed since it is not needed to prove the most basic facts about Euclidean domains (see below). However, this inequality can always be arranged to occur by changing the choice of ''v'', as follows: if (''D'',''v'') is a Euclidean domain as given above then the function ''w'' defined on nonzero elements of ''D'' by ''w''(''a'') = least value of ''v''(''ax'') as ''x'' runs over nonzero elements of ''D'' also makes ''D'' a Euclidean domain according to the above definition and it satisfies ''w''(''ab'') ≥ ''w''(''a'') for all nonzero ''a'' and ''b'' in ''D''. Examples of Euclidean domains include: |
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