Balazs Komuves writes:
It seems to me that a typical Euclidean domain does not have any kind of meaningful canonical associate / unit map. Examples:
- The Gaussian integers Z[i] (units are 1,-1,i,-i; what would be the associated element of 5+7i ?)
Prelude Data.Algebra Data.Complex> associate (5 :+ 7) :: Complex Integer 5 :+ 7 Prelude Data.Algebra Data.Complex> unit (5 :+ 7) :: Complex Integer 1 :+ 0 Prelude Data.Algebra Data.Complex> associate (5 :+ (-7)) :: Complex Integer 7 :+ 5 Prelude Data.Algebra Data.Complex> unit (5 :+ (-7)) :: Complex Integer 0 :+ (-1) Doing it by quadrants seems a reasonable choice. It's arbitrary to some extent, but then so are div and mod on Gaussian integers. But when we write a function (e.g. gcd), we have to provide _an_ answer, even it's not distinguished in some obvious way.