[GAP Forum] Re: Real field

[Alexander Hulpke]

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,
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