Opened 12 years ago

Last modified 11 years ago

#10771 closed enhancement

gcd and lcm for fraction fields — at Initial Version

Reported by: SimonKing Owned by: AlexGhitza
Priority: major Milestone: sage-4.7.2
Component: basic arithmetic Keywords: gcd lcm fraction fields
Cc: burcin Merged in:
Authors: Reviewers:
Report Upstream: N/A Work issues:
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Description

At sage-devel, the question was raised whether it really is a good idea that the gcd in the rational field should return either 0 or 1.

Since any non-zero element of QQ qualifies as gcd of two non-zero rationals, it should be possible to define gcd and lcm, so that gcd(x,y)*lcm(x,y)==x*y holds for any rational numbers x,y, and so that gcd(QQ(m),QQ(n))==gcd(m,n) and lcm(QQ(m),QQ(n))==lcm(m,n) for any two integers m,n.

Moreover, it should be possible to provide gcd/lcm for any fraction field of a PID: Note that currently gcd raises a type error for elements of Frac(QQ['x']).

The aim is to implement gcd and lcm as ElementMethods of the category QuotientFields().

It seems that defining gcd(a/b,c/d) = gcd(a,c)/lcm(b,d) and lcm(a/b,c/d) = lcm(a,c)/gcd(b,d) works, under the assumption that a/b and c/d are reduced fractions. Note: Since we need gcd for a,b,c,d anyway, it is no problem to reduce the fractions.

But I am not 100% sure whether that approach is mathematically sober.

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