Fix membership in QQbar for number field elements -- canonical embedding of subfields

[simon-king-jena]

Reported by Alex Raichev
on sage-support.

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.

CC: @kliem @slel

Component: algebra

Keywords: canonical embedding subfield

Issue created by migration from https://trac.sagemath.org/ticket/4621

[robertwb]

Attachment: 4621-qqbar-embed.patch.gz

[robertwb]

comment:1

This issue is fixed. Followup about embedding into QQbar at #7960.

[wjp]

comment:3

A side effect of this patch is the following because it now tries to explicitly convert its argument to QQ. Is that desirable?

sage: GF(7)(2) in QQbar
True

(Related 'in's:

sage: GF(7)(2) in ZZ
True
sage: GF(7)(2) in QQ
False
sage: GF(7)(2) in QQbar
True

)

[robertwb]

comment:4

This exposes a separate but that == for QQbar is not symetric...

sage: GF(7)(2) == QQbar(2)
False
sage: QQbar(2) == GF(7)(2)
True # after the patch, BOOM before, should be False

[robertwb]

comment:5

See #7984 for a fix for QQbar cmp.

[robertwb]

comment:6

For this to return True, one would have to change the definition of canonical comparison--not something that should be done lightly.

[mwhansen]

comment:7

Just a note -- this patch no longer works with #7984.

[videlec]

comment:12

The reason why this fails is

sage: F.<a>= NumberField(x^2-2)
sage: two = F(2)
sage: QQbar(two)
Traceback (most recent call last):
...
TypeError: Illegal initializer for algebraic number

One way to fix it is to be more flexible on creation of algebraic number (in AA._element_constructor_ or QQbar._element_constructor_) or to implement a method _algebraic_ to number field elements.

Vincent

[videlec]

comment:13

Replying to @videlec:

The reason why this fails is

sage: F.<a>= NumberField(x^2-2)
sage: two = F(2)
sage: QQbar(two)
Traceback (most recent call last):
...
TypeError: Illegal initializer for algebraic number

One way to fix it is to be more flexible on creation of algebraic number (in AA._element_constructor_ or QQbar._element_constructor_) or to implement a method _algebraic_ to number field elements.

the above is fixed in #14485 and #20400 but it does not solve the containment test!

[slel]

comment:16

To put it another way.

In Sage 9.3.rc0:

sage: root2 = QuadraticField(2).gen()
sage: root2 in QQbar, root2^2 in QQbar  # expected: (True, True)
(False, False)

[videlec]

comment:17

Replying to @slel:

To put it another way.

In Sage 9.3.rc0:

sage: root2 = QuadraticField(2).gen()
sage: root2 in QQbar, root2^2 in QQbar  # expected: (True, True)
(False, False)

The embedding is not set by writing only QuadraticField(2) (see also #30518 comment:10). You can compare with

sage: root2 = QuadraticField(2, embedding=AA(2).sqrt()).gen()
sage: root2 in QQbar, root2^2 in QQbar
(True, True)

Though the following is definitely annoying

sage: QuadraticField(2).one() in QQbar  # would better be true
False

[videlec]

comment:18

Note that it will quickly become annoying with extensions

sage: K.<a> = QuadraticField(2, embedding=AA(2).sqrt())
sage: x = polygen(ZZ)
sage: L.<b> = K.extension(x**3 - x**2 - x - 1) 

• Do you expect QQbar(L(a)) to work?
• What should be the result of L(a) in QQbar?

[videlec]

comment:19

Note also that 1 == 1 does not hold in the following situation

sage: QuadraticField(2).one() == QuadraticField(3).one()
True

Again, with properly set embeddings it compares as a user might expect

sage: K2 = QuadraticField(2, embedding=AA(2).sqrt())
sage: K3 = QuadraticField(3, embedding=AA(3).sqrt())
sage: K2.one() == K3.one()
True

[videlec]

comment:20

The only way I can imagine a fix would be to implement intersection of parents as part of the coercion model. It would have at least the following requirements

• C = A.intersection(B) is a parent with injective coercions in both A and B
• A.intersection(B) is identical to B.intersection(A)

Given that, we could design a more reasonable sage.structure.element.Element.__richcmp__.

[slel]

comment:21

Thanks for launching a discussion on sage-devel:

• https://groups.google.com/g/sage-devel/c/7-C6cpUyJdI

[mkoeppe]

comment:22

Sage development has entered the release candidate phase for 9.3. Setting a new milestone for this ticket based on a cursory review.

[mkoeppe]

comment:23

Setting a new milestone for this ticket based on a cursory review.

[mkoeppe]

comment:24

Stalled in needs_review or needs_info; likely won't make it into Sage 9.5.