embedding of number field element encapseled in symbolic ring

[Daniel Krenn]

I want to take an number field element embedded in the symbolic ring
like the imaginary I and add it to an algebraic number:
sage: I.pyobject() + QQbar(sqrt(2))
This results in
TypeError: unsupported operand parent(s) for '+': 'Number Field in I
with defining polynomial x^2 + 1' and 'Algebraic Field'
as there is no *canonical* coercion as no embedding of the number field
is specified.

How can I specify this embedding such that it is used e.g. for the
symbolic I?

I've tried
sage: F = NumberField(x^2 + 1, 'i', embedding=QQbar(I))
which makes
sage: F.0 + QQbar(sqrt(2))
1.414213562373095? + 1*I
work. However,
sage: SR(F.0)
I
so it is converted to the existing I and the example at the top
sage: SR(F.0).pyobject() + QQbar(sqrt(2))
fails again.

Daniel

[Daniel Krenn]

On 2017-01-05 10:55, Daniel Krenn wrote:
> as there is no *canonical* coercion as no embedding of the number field
> is specified.
>
> How can I specify this embedding such that it is used e.g. for the
> symbolic I?

This looks weird: I is defined in sage.libs.pynac.pynac via
K = QuadraticField(-1, 'I', embedding=CC.gen(), latex_name='i')
pynac_I = K.gen()
so there is an embedding defined, but QQbar this seems to be not
sufficient to recognize it as an embedding into QQbar.

How can this be fixed?

[Daniel Krenn]

It seems like the embedding is simply ignored in some sense:
sage: QQbar(QuadraticField(-1, 'I', embedding=-CC.gen()).gen())
I
whereas
sage: CC(QuadraticField(-1, 'I', embedding=-CC.gen()).gen())
-1.00000000000000*I

[Dima Pasechnik]

to me this looks an unimplemented feature of SR; basically, you want SR to be able to manipulate

algebraic numbers with a specified embedding in the same ring with transcendental functions.

(perhaps Mathematica does support this, although I'm not 100% sure)

[Vincent Delecroix]

Le 05/01/2017 à 11:33, Daniel Krenn a écrit :
> On 2017-01-05 11:29, Daniel Krenn wrote:
>> On 2017-01-05 10:55, Daniel Krenn wrote:
>>> as there is no *canonical* coercion as no embedding of the number field
>>> is specified.
>>>
>>> How can I specify this embedding such that it is used e.g. for the
>>> symbolic I?
>>
>> This looks weird: I is defined in sage.libs.pynac.pynac via
>> K = QuadraticField(-1, 'I', embedding=CC.gen(), latex_name='i')
>> pynac_I = K.gen()
>> so there is an embedding defined, but QQbar this seems to be not
>> sufficient to recognize it as an embedding into QQbar.
>
> It seems like the embedding is simply ignored in some sense:
> sage: QQbar(QuadraticField(-1, 'I', embedding=-CC.gen()).gen())
> I

To my mind, this one should actually raise an error. The proper way to
declare an embedding in QQbar is via

sage: K = QuadraticField(-1, 'I', embedding=-QQbar(I))
sage: K.0
I
sage: QQbar(K.0)
-I

which works perfectly fine.

Vincent

[Daniel Krenn]

On 2017-01-07 11:09, Vincent Delecroix wrote:
> Le 05/01/2017 à 11:33, Daniel Krenn a écrit :
>> It seems like the embedding is simply ignored in some sense:
>> sage: QQbar(QuadraticField(-1, 'I', embedding=-CC.gen()).gen())
>> I
>
> To my mind, this one should actually raise an error.

Ok.

> The proper way to
> declare an embedding in QQbar is via
>
> sage: K = QuadraticField(-1, 'I', embedding=-QQbar(I))
> sage: K.0
> I
> sage: QQbar(K.0)
> -I
>
> which works perfectly fine.

Is it possible to define one embedding from a number field into CC and
another into QQbar (at "the same time")?

Daniel

[Nils Bruin]

On Sunday, January 8, 2017 at 12:32:51 AM UTC-8, Daniel Krenn wrote:

Is it possible to define one embedding from a number field into CC and
another into QQbar (at "the same time")?

No. A field only has one slot for an embedding. This restriction doesn't have good technical grounds for existing. It does make it a tiny bit harder to introduce inconsistencies in the coercion graph, though.

If a field embeds into QQbar, then there should be an implied coercion into CC, though: QQbar as implemented in sage is a subfield of the field of complex numbers (as you can see from how the elements print). And indeed this works:

sage: K.<a>=NumberField(x^3-2,embedding=QQbar(2^(1/3)))
sage: a+CC(1)
2.25992104989487

[Marc Mezzarobba]

Nils Bruin wrote:
> If a field embeds into QQbar, then there should be an implied coercion
> into CC, though: QQbar as implemented in sage is a subfield of the
> field of complex numbers (as you can see from how the elements print).

Let me add that there are long-standing tickets (#5355, #12715) for
making number field embeddings go into AA/QQbar instead of RR/CC. Quite
a bit of progress has been made towards that goal, but we're not quite
there yet.

--
Marc