On Thu, Feb 10, 2011 at 9:49 AM, Simon King <simon.k...@uni-jena.de> wrote: > Hi Luis, > > On 10 Feb., 17:48, luisfe <lftab...@yahoo.es> wrote: >> ... >> You could have both consistencies. That depends on how you define gcd >> and lcm: >> >> - Quotient fields as described by Bruno. >> - Fields: zero if both elements are zero. A non-zero element >> otherwise (most fields would choose 1 here). >> - PID: a generator of the corresponding ideal. >> >> This is not trivial. For instance Fields do not have a default gcd/lcm >> method. I asked in sage-devel about some time ago about sensible >> approaches >> here.http://groups.google.com/group/sage-devel/browse_thread/thread/12524b... > > Yes, on my way home, I thought that perhaps "lcm(4/1,2)=1" is not so > obvious as I first found. lcm(a,b) has to be a generator of the > intersection of the ideals generated by a and b. Of course, 1 is a > quite canonical generator for the only non-zero ideal in a field -- > simply because any field has a 1. > > But any other non-zero element is fine as well, in a field. So, after > all, defining lcm(a/b,c/d)=lcm(a,c)/gcd(b,d) for fraction fields of > principal ideal domains makes more sense than I originally thought. > And with gcd(a/b,c/d)=gcd(a,c)/lcm(b,d), we would indeed have > gcd(x,y)*lcm(x,y)=x*y. > > According to Richard Fateman, that definition seems to be be used in > Maxima ("in maxima, gcd(1/4,1/6) is 1/12, lcm is 1/2"). But > according to Tim Daly, Axiom returns 1 as lcm of any two rationals. > So, should Sage stay on the side of Axiom or switch to the side of > Maxima?
It should switch to the side of Maxima/Pari/Mathematica/etc. in this. -- William -- William Stein Professor of Mathematics University of Washington http://wstein.org -- 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
