> Well, I used to use gcd for obtaining the primitive integral vector > with a specified rational direction. My concern on Trac 3214 was that > gcd(a1, ..., ak) depended on the order of arguments and I wanted it to > be fixed. The eventual solution was to agree that gcd as the "greatest > common divisor" does not really make much sense for fields, but > instead of raising an exception it can just return 1. This meant that > I cannot use it for my original purpose (not a big deal - it is easy > to do more directly), but I think that it was quite a sensible > decision: > 1) I don't recall seeing gcd of rational numbers in any book or paper > 2) there is clearly no natural extension of this notion from ZZ to QQ > 3) the name itself indeed is very strange applied to fields. > > So I personally think that any kind of gcd/lcm combinations of > numerators/denominators of rational numbers should have some other > more appropriate names, since making up some conventions for gcd is > potentially dangerous and may make code using it harder to understand > if a reader thinks of gcd differently than the original author... >
I agree that it's a little bit awkward, but I think that we should go with the maxima/pari/mathematica convention. * The issue the OP brought up, where gcd silently gave nonsensical issues when one of the "integers" accidentally got divided by 1 (or you used / instead of // in a case where you know one is divisible by the other) is compelling. * The gcd should be a generator of the fractional ideal generated by the inputs; the lcm of the intersection as long as that fractional ideal is principal (if we're in a non-PID or fraction field of a non-PID I think we should raise an error). There's an ambiguity about units; we should make a consistent choice between the PID and its fraction field. In the case of integers and rationals, this means we choose the results to be positive and have the relation gcd(x,y)*lcm(x,y) = abs(x,y) as John pointed out. David -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
