- We know ,euclidean algorthm is feasible in the set of integers.
- Taking motivation from this,we define an Euclidean Domain(E.D.) as follows:
R is an E.D. if it is a domain,andwe have a map d:R->non-zero integers,
such that, for given a,b(non-zero),
there exists q,r with
a=bq+r and d(q)
Anonymous Anonymi
Christian Neurohr
But,Z is an ED too,with d(a)=|a|.but,|a+b| \la max{ |a|,|b| }does not hold in general.
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Anna Limbach
The uniqueness of (q,r) in Z is only valid, when requiring r to be non-negative. This can't be expected in general, since not every Euclidean Domain is ordered.
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Róbert Freud
The quotient and remainder are not unique in the integers, because the Euclidean function is the absolute value, i.e. a=bq+r and |r|
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