#4621: '2' not in QQbar -- canonical embedding of subfields
-----------------------+----------------------------------------------------
Reporter: SimonKing | Owner: tbd
Type: defect | Status: new
Priority: major | Milestone: sage-3.3
Component: algebra | Keywords: canonical embedding subfield
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Reported by Alex Raichev at http://groups.google.com/group/sage-
support/browse_thread/thread/c11289d794299903
{{{
sage: F.<a>= NumberField(x^2-2)
sage: a^2
2
sage: a^2 in QQ
True
sage: a^2 in QQbar
False
sage: 2 in QQbar
True
}}}
or more directly
{{{
sage: F(2) in QQbar
False
}}}
Perhaps related to this is
{{{
sage: F.<a>=NumberField(x^2-2)
sage: QQ.is_subring(F)
True
sage: F.is_subring(QQbar)
False
}}}
Robert Bradshow comments that `F.is_subring(QQbar)` should be `False`,
because `QQbar` has a canonical embedding into `CC`, but `F` has not.
So, from that point of view, it makes sense that `a^2` is in `F` but not
in `QQbar`. However, `a^2` is equal to `2` after all, and hence is in a
part of `F` that ''does'' have a canonical embedding.
In other words, we have a field element x in F_1 such that there is in
fact a subfield F_2 of F_1 with x in F_1. Moreover, we have a field F_3
such that F_2 has a canonical embedding into F_3, but F_1 has no canonical
embedding.
Is it possible for Sage to detect that situation?
Idea: Is there a ''unique'' maximal subfield F_m of F_1 that has a
canonical embedding into F_3? If there is, there could be a method
`max_subfield_coercing_into(...)`.
Then, in the original example, we probably have
{{{
sage: F.max_subfield_coercing_into(QQbar)
Rational Field
}}}
and then `x in QQbar` would answer True, since
{{{
sage: x in F_1.max_subfield_coercing_into(QQbar)
True
}}}
Sorry if that idea is not realistic.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/4621>
Sage <http://sagemath.org/>
Sage - Open Source Mathematical Software: Building the Car Instead of
Reinventing the Wheel
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