2015-01-25 22:07 UTC+01:00, John Cremona <john.crem...@gmail.com>:
>
> 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.
Agreed!
> I acknowledge the fact that Z[x] is a UFD so that there is a
> definition of gcd which makes sense here, but it not the ideal
> generator of the other definition and hence there is no extended gcd,
> i.e. no Bezout identity.
Agreed!
> I seem to have argued the case that in a UFD which is no a PID, it
> makes sense to define gcd(a,b) but not xgcd(a,b).
Sage does propose a definition
(sage.rings.polynomials.polynomial_integer_dense_flint):
def xgcd(self, other):
This function can't in general return ``(g,s,t)`` as above,
since they need not exist. Instead, over the integers, we
first multiply `g` by a divisor of the resultant of `a/g` and
`b/g`, up to sign, and return ``g, u, v`` such that
``g = s*self + s*right``. But note that this `g` may be a
multiple of the gcd.
If ``self`` and ``right`` are coprime as polynomials over the
rationals, then ``g`` is guaranteed to be the resultant of
self and right, as a constant polynomial.
Beyond the first sentence which refers to an "above", this is not very
clear to me. Perhaps xgcd is not standard (or a bad standard)?
Vincent
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