#8327: Implement the universal cyclotomic field, using Zumbroich basis
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Reporter: nthiery | Owner:
Type: enhancement | Status:
needs_review
Priority: major | Milestone: sage-5.6
Component: number fields | Resolution:
Keywords: Cyclotomic field, Zumbroich basis | Work issues:
Report Upstream: N/A | Reviewers:
Authors: Christian Stump, Simon King | Merged in:
Dependencies: #13727, #13728 | Stopgaps:
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Comment (by chapoton):
> > - is_unit should be removed, as already implemented in the class of
Field elements
>
> the inherited {{{is_unit}}} method is not capturing the situation. It's
implementation is
> {{{
> def is_unit(self):
> if self == 1 or self == -1:
> return True
> raise NotImplementedError
> }}}
I do not understand : if one look at Ticket #13728, the method is_unit
does check that an element is not zero !
> > - maybe consider a coercion of real elements to the field AA ?
>
> This is impossible, isn't it? Here is a quote from the documentation:
>
> "Another consequence of the consistency condition is that coercions can
only go from exact rings (e.g., the rationals QQ) to inexact rings (e.g.,
real numbers with a fixed precision RR), but not the other way around."
>
> Or am I misunderstanding something?
Well, I do not know. The point is that we can currently use QQbar(z) for z
in UCF. I wonder wether one could do AA(z) for z in UCF and real, because
AA is just the set of real elements of QQbar. But maybe I do not
understand something..
{{{
sage: toto=E(6)
sage: QQbar(toto)
0.500000000000000? + 0.866025403784439?*I
sage: riri=QQbar(toto)
sage: riri in AA
False
sage: riri+riri.conjugate() in AA
True
sage: QQbar(toto+toto.conjugate()) in AA
True
sage: AA(riri+riri.conjugate()).parent()
Algebraic Real Field
}}}
> > 1 the field UCF is contained in the complex field
>
> This coercion is going through QQbar, so I only documented the later.
Yes, But I rather meant the mathematical point : the field is defined as
an embedded field, not as an abstract field..
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8327#comment:117>
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