Extract roots in NumberField if possible

[BrentBaccala]

NumberField previously evaluated integral powers in the NumberField, and evaluated all fractional powers in the symbolic ring.

This patch makes NumberField attempt to evaluate the fractional power within the field, and only falls back on the symbolic ring if this fails.

There's a few interesting changes in the test suite.

Old code:

sage: QQbar((2*I)^(1/2))
1 + 1*I
sage: (2*I)^(1/2)
sqrt(2*I)
sage: I^(2/3)
I^(2/3)

New code:

sage: QQbar((2*I)^(1/2))
I + 1
sage: (2*I)^(1/2)
I + 1
sage: I^(2/3)
-1

The first change is just cosmetic. The second makes good sense, as Sage is now evaluating an expression it didn't before. The third change is more troubling.

The explanation lies in the definition of I:

sage: I.parent()
Symbolic Ring
sage: I.pyobject().parent()
Number Field in I with defining polynomial x^2 + 1

In this number field, there is a single cube root of I (-I). Squaring -I gives us -1, so I^(2/3)=-1.

My opinion is that the new behavior of NumberField is correct and preferred, but perhaps I should be defined in QQbar, not in a NumberField.

CC: @slel

Component: algebra

Keywords: NumberField

Author: Brent Baccala

Branch: 17b93d6

Reviewer: Sébastien Labbé

Issue created by migration from https://trac.sagemath.org/ticket/25218

[BrentBaccala]

Author: Brent Baccala

[BrentBaccala]

New commits:

75a46a0	Trac #25218: NumberField attempts to evaluate fractional powers
087174b	Trac #25218: update test cases altered by this patch

[BrentBaccala]

Commit: 087174b

[BrentBaccala]

Branch: u/gh-BrentBaccala/25218

[slel]

comment:2

Regarding the suggestion that

perhaps I should be defined in QQbar, not in a NumberField,

see also #18036.

[slel]

comment:3

The following tickets are possibly related.

• Gaussian and Eisenstein integers #7545: Gaussian and Eisenstein integers
  Gaussian and Eisenstein integers #7545

• I.parent() should not be the symbolic ring #18036: I.parent() should not be the symbolic ring
  I.parent() should not be the symbolic ring #18036

• Conversion from SR to number fields #22208: Conversion from SR to number fields
  Conversion from SR to number fields #22208

• SR('I') is not consistent #25178: SR('I') is not consistent
  SR('I') is not consistent #25178

• Extract roots in NumberField if possible #25218: Extract roots in NumberField if possible
  Extract roots in NumberField if possible #25218

[sagetrac-git]

Changed commit from 087174b to 592e263

[sagetrac-git]

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

592e263

Trac #25218: fix typo

[seblabbe]

Reviewer: Sébastien Labbé

[seblabbe]

Changed commit from 592e263 to 17b93d6

[seblabbe]

comment:6

I did small spaces fixes. If you agree with my changes, please change the status to positive review.

---

New commits:

17b93d6

25218: space fixes

[seblabbe]

Changed branch from u/gh-BrentBaccala/25218 to u/slabbe/25218

[vbraun]

Changed branch from u/slabbe/25218 to 17b93d6

[mezzarobba]

Replying to @BrentBaccala:

The third change is more troubling.

The explanation lies in the definition of I:

sage: I.parent()
Symbolic Ring
sage: I.pyobject().parent()
Number Field in I with defining polynomial x^2 + 1

In this number field, there is a single cube root of I (-I). Squaring -I gives us -1, so I^(2/3)=-1.

My opinion is that the new behavior of NumberField is correct and preferred, but perhaps I should be defined in QQbar, not in a NumberField.

I think this is very unfortunate. In Sage, QQ[i] automatically comes with a complex embedding, for which i^2/3 (= exp(iπ/3)) is perfectly well defined. It is really confusing to have a basic operator like ^ (whose evaluation normally uses the coercion framework) that is incompatible with the embedding.

See #30783.

[mezzarobba]

Changed commit from 17b93d6 to none