On October 17, 2011 19:19:18 Paterson, Ross wrote:
Balazs Komuves writes:
Rings with unity have a canonical map, actually a ring homomorphism (but not necessarily injection) from the integers, namely for the natural integer N, you add together the unit element with itself N times. For negative N, you take the additive inverse.
For fields, you would try to extend this to rationals; however, it seems that because of the non-injectivity of the above, this won't always work. Example: finite fields. In a finite field of order P, we would have f(N/P) = f(N)/f(P) = f(N)/0 which is not defined.
Good point. Mind you we already have this with Ratio Int and friends.
I was also wondering about the EuclidanDomain "associate" and "unit" functions. YAP requires, for all x x = associated x * unit x (where unit x has an multiplicative inverse) associate (unit x) = unit (associate x) = 1 http://hackage.haskell.org/packages/archive/yap/0.1/doc/html/Data-YAP- Algebra.html#t:EuclideanDomain This seem very much like generalized absolute and signum functions. The wikipedia page on Euclidean Domains, however, doesn't mention such functions. Rather it works with the concept of a "euclidian" function which gives a mapping onto the non-negative integers such that euclidean (mod a b) < euclidean b whenever mod a b /= 0 http://en.wikipedia.org/wiki/Euclidean_domain#Definition Are these two viewpoints connected? Does a Euclidian domain's euclidian function somehow give rise to the "associate" and "unit" functions? Thanks! -Tyson