Generalize pushout of number fields with compatible embeddings

[gh-kliem]

The pushout of two algebraic number fields with a coerce embedding in AA (the real algebraic fields) is implemented as being AA. As a consequence, it is possible to add elements from different number fields

sage: K.<a> = NumberField(x^3 - 2, embedding=AA(2)**(1/3))
sage: L.<b> = NumberField(x^2 - 3, embedding=AA(3)**(1/2))
sage: a + b
2.991971857463751?

Though, the pushout is not implemented for fields with coerce embedding in QQbar (algebraic field).

sage: K.<a> = NumberField(x^3 - 2, embedding=QQbar(2)**(1/3) * QQbar.zeta(3))
sage: L.<b> = NumberField(x^3 - 3, embedding=QQbar(3)**(1/3) * QQbar.zeta(3)**2)
sage: a + b
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +

This ticket implements it.

[gh-kliem]

New commits:

​aac8e0a

implement pushout for generic number fields with compatible embeddings

[vdelecroix]

A pushout is already available

sage: K.<a> = NumberField(x^2 - 2, embedding=AA(2).sqrt())
sage: L.<b> = NumberField(x^2 - 3, embedding=AA(3).sqrt())
sage: a+b
3.146264369941973?
sage: K._pushout_(L)
Algebraic Real Field

(was implemented in #20746)

Going via composite field is very fragile (order dependent, non canonical, expensive, etc).

[gh-kliem]

Thanks for pointing this out. I was aware of this method, but not that you had the discussion before.

Nevertheless, one could just devote this ticket here to implementing the case, where the fields are embedded into QQbar.

I figured this would be useful for rings in general.

[gh-mwageringel]

Replying to gh-kliem:

Nevertheless, one could just devote this ticket here to implementing the case, where the fields are embedded into QQbar.

I figured this would be useful for rings in general.

Assuming this also makes coercion between AA and number fields embedded in QQbar possible, this would then help solve #28489 as far as I can see (possibly after #28774).

[vdelecroix]

description updated then

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​d3258e6

pushout for number fields with compatible embeddings

[gh-kliem]

I also fixed a bug of the previous construction. It did try to coerce number fields embedded into RLF into AA, which did not work (the attribute _embedded_real is set to True in that case).

There is a failing doctest now in sage/rings/qqbar.py line 167, for which I need some help.

The test wants to make sure, that QQ[I] and QQbar cannot be implicitly coerced, as QQ[I] does not come with an embedding.

This makes sense, however this is not the way it is implemented:

sage: QQ[I]
Number Field in I with defining polynomial x^2 + 1 with I = 1*I

So QQ[I] does come with an embedding to QQbar and therefore I think my construction is correct in using this embedding. Unlike this example

sage: K.<a> = NumberField(x^2 + 1); K
Number Field in a with defining polynomial x^2 + 1

where my construction does not give a pushout.

[gh-kliem]

To be more precise, as it is QQ[I] comes with an embedding to complex lazy field:

sage: coercion_model.explain(QQ[I], CLF)
Coercion on left operand via
    Generic morphism:
      From: Number Field in I with defining polynomial x^2 + 1 with I = 1*I
      To:   Complex Lazy Field
      Defn: I -> 1*I
Arithmetic performed after coercions.
Result lives in Complex Lazy Field
Complex Lazy Field

and there is a coercion from QQbar to CLF

sage: coercion_model.explain(QQbar, CLF)
Coercion on left operand via
    Generic morphism:
      From: Algebraic Field
      To:   Complex Lazy Field
Arithmetic performed after coercions.
Result lives in Complex Lazy Field
Complex Lazy Field

I find it very confusing, that there is a doctest fixing that there shall be no coercion of QQ[I] and CLF. If this is the way we want it to be, there should not be coercion of QQ[I] and CLF in the first place.

The same holds for QQ[sqrt(2)] btw. There is no embedding of QQ[sqrt(2), sqrt(3)] to RLF, but probably not by design but only because relative number fields do not support embeddings.

[gh-kliem]

Please tell me, if I should post a query on sage-devel instead.

[gh-kliem]

Ok, I just went back to just doing QQbar and added the case that both are embedded into the same field.

---

New commits:

​b4b2cf1

pushout for number fields with compatible embeddings

[vdelecroix]

Your example

sage: K.<sqrt2_plus_sqrt3> = NumberField(x^4 - 10*x^2 + 1)
sage: K2.<sqrt2> = NumberField(x^2-2, embedding=(sqrt2_plus_sqrt3^3 - 9*sqrt2_plus_sqrt3)/2)
sage: K3.<sqrt3> = NumberField(x^2-3, embedding=-(sqrt2_plus_sqrt3^3 - 11*sqrt2_plus_sqrt3)/2)
sage: sqrt2 + sqrt3
sqrt2_plus_sqrt3

is working without any modification to the _pushout_ method. You should simply remove what you wrote for number fields unless there is a real application for it.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​d46d6bb

remove unneeded test

[gh-kliem]

Ok. I see.

This example is handled by the _pushout_ method now, but I don't think it hurts. I wasn't aware that a canonical coercion is already discovered. Nice.

I could also just handle AA and QQbar, if this seems more reasonable.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​9a1bce3

small mistake in docstests

[vdelecroix]

Replying to gh-kliem:

Ok. I see.

This example is handled by the _pushout_ method now, but I don't think it hurts. I wasn't aware that a canonical coercion is already discovered. Nice.

I could also just handle AA and QQbar, if this seems more reasonable.

Yes, it would be better.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​8e70251

defining the pushout for number fields just for AA and QQbar

[embray]

Ticket retargeted after milestone closed

[mkoeppe]

Batch modifying tickets that will likely not be ready for 9.1, based on a review of the ticket title, branch/review status, and last modification date.

[vdelecroix]

Maybe we could be more greedy with something like

if codomain_self is not self or codomain_other is not other:
    try:
        return coercion_model.common_parent(codomain_self, codomain_other)
    except TypeError:
        pass

[mkoeppe]

Looks like a good improvement as is.

Feel free to open the suggestion from comment 21 as another ticket.

[gh-kliem]

Thank you.