Dear Forum, Dear Mathieu,
From the various comments I understand that it is not reasonable
to try to use the existing cyclotomic fields but that the "right"
solution would be to redefine from scratch the new field Q(Sqrt(5)).
Gap offers cyclotomic fields, finite fields, number fields, various
kind of rings, but not real fields. Is there an intrisic obstacle
to it?
GAP offers subfields of the cyclotomics (and thus real subfields of
the cyclotomics). The problem is simply with the comparison:
GAP needs to have a comparison (via `<`) defined on all of the
cyclotomics for example to form sets. at the moment this comparison
is rather ad-hoc and for irrational real numbers is not compatible
with the ordering induced by the real numbers.
This is indeed a pity, however doing so would be rather hard:
It is not sufficient to have a comparison only for some subfield, but
a comparison must be possible for any elements. Leaving out non-real
elements this means we have to be able to compare real elements of CF
(5), say, with elements of CF(317). The only method for this which I
am aware of is via numerical approximation, which is potentially very
expensive.
If anybody has a better idea of how to define a total order on the
cyclotomics that for the real subfield is compatible with the natural
ordering I'd be very happy to hear about.
Best,
Alexander
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: [EMAIL PROTECTED], Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
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