QQbar bug
Alex Raichev
I think i found a small bug relating to QQbar, the algebraic closure
of the rationals.
------------------------------------------------------------------------------------
| Sage Version 3.2, Release Date:
2008-11-20 |
| Type notebook() for the GUI, and license() for information. |
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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
Alex
Simon King
On Nov 26, 1:51 am, Alex Raichev <tortoise.s...@gmail.com> wrote:
> 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
sage: F.<a>=NumberField(x^2-2)
sage: QQ.is_subring(F)
True
sage: F.is_subring(QQbar)
False
Cheers,
Simon
mabshoff
please open a ticket. I would guess as you did that those two related.
Hopefully Carl will have some insight what is going on there.
Cheers,
Michael
Robert Bradshaw
This one should be false as QQbar comes with an embedding into CC,
but F does not (in other words, there is no canonical embedding of F
into QQbar). The a^2 in QQbar is a bug though.
- Robert
Simon King
On Nov 26, 11:34 am, mabshoff <Michael.Absh...@mathematik.uni-
Done, it is # 4621.
By the way, the above problem appears even more directly:
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. 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.
Best regards,
Simon
Robert Bradshaw
On Nov 26, 2008, at 3:30 AM, Simon King wrote:
>
> Dear Michael,
>
> On Nov 26, 11:34 am, mabshoff <Michael.Absh...@mathematik.uni-
> 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.
- Robert
John Cremona
On Nov 26, 8:31 pm, Robert Bradshaw <rober...@math.washington.edu>
wrote:
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
Robert Bradshaw
Yep. Note that once that number field coercion code gets merged,
QuadraticField(2) will have a preferred embedding into CC, and hence
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)
Yep. One just needs to fix the __call__ method of QQbar.
- Robert