#12715: Number field embeddings should go via AA and QQbar
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Reporter: davidloeffler | Owner: davidloeffler
Type: defect | Status: new
Priority: major | Milestone: sage-5.3
Component: number fields | Resolution:
Keywords: qqbar coercion | Work issues:
Report Upstream: N/A | Reviewers:
Authors: | Merged in:
Dependencies: | Stopgaps:
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Comment (by robharron):
Just wanted to point out that actually it is possible, though slightly
roundabout, to create such embeddings.
{{{
sage: x = polygen(QQ)
sage: f = x^3 - x -1
sage: r = f.roots(AA)[0][0]
sage: K.coerce_embedding()
Generic morphism:
From: Number Field in a with defining polynomial x^3 - x - 1
To: Algebraic Real Field
Defn: a -> 1.324717957244746?
}}}
If you look at the code that produces the embedding
(create_embedding_from_approx(K, gen_image) in number_field_morphisms.pyx)
you see that it chooses a lazy embedding if gen_image.parent() is not
exact (which is not the case here) *or* gen_image is not a root of the
defining polynomial of K. The latter is the case in your example. Perhaps
sage is calling you lazy for passing it an approximation to the root!
Joking aside I think it would be a capital idea to have AA as the codomain
rather than RLF. One way to accomplish this would be to take replace
gen_name with AA(gen_name.exact_rational()) (and analogously for the real
and imaginary parts if gen_name is complex) and replace RLF with AA. Sound
good?
It would probably also make sense to do this with the Minkowski_embedding
function (though maybe that would follow from this change?).
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12715#comment:2>
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