2015-01-25 23:08 UTC+01:00, Jeroen Demeyer <jdeme...@cage.ugent.be>:
> On 2015-01-25 22:07, John Cremona wrote:
>> I would say that this is undefined since the ideal generated by x+2
>> and x+4 is (2,x) which is not principal. I think that the most useful
>> definition of gcd is a generator of the ideal generated by the two
>> objects *if* that ideal is principal.
> I disagree. There are rings where a gcd can be defined(*) even if the
> ideal is not principal. Multi-variate polynomial rings are a very
> natural example. I see no reason to disallow gcd() there.
>
> xgcd() is a different story, there you really need a stronger condition
> on the ring, namely you need a Bézout domain (every finitely generated
> ideal is principal, which is slightly weaker than PID).
Nice. Does Sage knows something about the existence Bézout domain? We
should definitely have one!
> So I think that gcd() should stay as it is and xgcd() should raise
> ArithmeticError if the ideal is not principal.
>
> (*) literally as greatest common divisor, where "greatest" is taken
> using the partial order defined by division.
So what is the proper name for the current xgcd implemented on
univariate polynomials over ZZ? It has been introduced long ago
commit beaec284
Author: Burcin Erocal <bur...@erocal.org>
Date: Mon Jun 30 11:56:56 2008 +0200
Make FLINT the default backend for ZZ[x], trac#2357.
The thing is that having a method xgcd on the elements automatically allow
sage: xgcd(p,q)
Vincent
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