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: | |

Branch: | Commit: | ||

Dependencies: | Stopgaps: |

### 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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