On Nov 26, 8:31 pm, Robert Bradshaw <[EMAIL PROTECTED]>
wrote:
> On Nov 26, 2008, at 3:30 AM, Simon King wrote:
>
>
>
>
>
> > Dear Michael,
>
> > On Nov 26, 11:34 am, mabshoff <[EMAIL PROTECTED]
> > dortmund.de> wrote:
> >> please open a ticket. I would guess as you did that those two
> >> related.
>
> > Done, it is # 4621.
>
> > By the way, the above problem appears even more directly:
> > sage: F.<a>= NumberField(x^2-2)
> > sage: 2 in QQbar
> > True
> > sage: F(2) in QQbar
> > False
>
> > Although F has no canonical embedding into QQbar, my understanding is
> > that there is a hopefully unique maximal subfield of F that does have
> > a canonical embedding into QQbar.
>
> Into the mathematical \bar{Q}, yet. Sage's QQbar is \bar{Q} with a
> choice of embedding into \C, and as F does not have a (chosen)
> embedding into \C it doesn't have a chosen embedding into QQbar.
>
> > If this is correct, there could be a
> > method F.max_subfield_coercing_into(QQbar), and since F(2) is in that
> > subfield, one has a reason to expect `F(2) in QQbar` to be True.
>
> One *does* expect F(2) to be in QQbar, the same that one expects the
> rational number 4/2 to be in ZZ, so I agree that the above is a bug.
>
sage: F.<a>=NumberField(x^2-2)
sage: F.embeddings(QQbar)
[
Ring morphism:
From: Number Field in a with defining polynomial x^2 - 2
To: Algebraic Field
Defn: a |--> -1.414213562373095?,
Ring morphism:
From: Number Field in a with defining polynomial x^2 - 2
To: Algebraic Field
Defn: a |--> 1.414213562373095?
]
Is the problem that there is more than one embedding of F into QQbar?
Of course, the image of 2 will be the same with each:
sage: twos = [f(2) for f in F.embeddings(QQbar)]
sage: twos
[2, 2]
sage: twos[0] == twos[1]
True
But note that this crashes horribly:
sage: twos[1] == 2
True
sage: twos[1] == F(2)
---------------------------------------------------------------------------
TypeError (etc)
> - Robert
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