number_field_elements_from_algebraics incorrectly simplify root of unity?

[user202729]

Steps To Reproduce

Run

L, [ww], phi = number_field_elements_from_algebraics([(-1)^(2/3)])
assert ww != 1

Expected Behavior

That ww become a primitive 3rd root of unity.

Actual Behavior

ww = 1.

Additional Information

No response

Environment

- **OS**: Linux
- **Sage Version**: SageMath version 10.1, Release Date: 2023-08-20

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide

[fchapoton]

Looks like a bug indeed. You could avoid using the symbolic ring:

sage: L, [ww], phi = number_field_elements_from_algebraics([E(3)**2])
sage: L, ww, phi
(Cyclotomic Field of order 3 and degree 2,
 -zeta3 - 1,
 Ring morphism:
   From: Cyclotomic Field of order 3 and degree 2
   To:   Algebraic Field
   Defn: zeta3 |--> -0.500000000000000? + 0.866025403784439?*I)

[DaveWitteMorris]

I think these are the source of the erroneous answer:

sage: AA((-1)^(2/3))
1
sage: from sage.symbolic.expression_conversions import algebraic
sage: algebraic((-1)^(2/3), AA)
1

Shouldn't they give an error, instead of converting?

[DaveWitteMorris]

Sorry, I removed the bug tag from the wrong ticket.

[DaveWitteMorris]

I think this is the problem:

In lines 105-110 of the method arithmetic of class AlgebraicConverter(Converter) (in src/sage/symbolic/expression_conversion/algebraic.py), we see that the expression AA((-1)^(2/3)) is calculated as (AA(-1))**(2/3). But this is not what we want, because the documentation of AA says "AlgebraicReal [is] closed under ... rational powers (including roots), except that for a negative AlgebraicReal, taking a power with an even denominator returns an AlgebraicNumber instead of an AlgebraicReal." This means that raising an element of AA to the 2/3 power will give a result that is in AA. That's not the answer that we want for this expression.

[mezzarobba]

See also #12074, #12745, #18333.

[kwankyu]

Looks like a bug indeed. You could avoid using the symbolic ring:

sage: L, [ww], phi = number_field_elements_from_algebraics([E(3)**2])
sage: L, ww, phi
(Cyclotomic Field of order 3 and degree 2,
 -zeta3 - 1,
 Ring morphism:
   From: Cyclotomic Field of order 3 and degree 2
   To:   Algebraic Field
   Defn: zeta3 |--> -0.500000000000000? + 0.866025403784439?*I)

Since

sage: QQbar((-1)^(1/3))
0.500000000000000? + 0.866025403784439?*I
sage: QQbar(E(3))
-0.500000000000000? + 0.866025403784439?*I
sage: QQbar((-1)^(2/3))
-0.500000000000000? + 0.866025403784439?*I
sage: QQbar(E(3)**2)
-0.500000000000000? - 0.866025403784439?*I

E(3)**2 is not equal to (-1)^(2/3) but E(3) is. Note that

sage: L, [ww], phi = number_field_elements_from_algebraics([QQbar((-1)^(2/3))])
sage: L
Cyclotomic Field of order 6 and degree 2
sage: ww
zeta6 - 1
sage: phi
Ring morphism:
  From: Cyclotomic Field of order 6 and degree 2
  To:   Algebraic Field
  Defn: zeta6 |--> 0.500000000000000? + 0.866025403784439?*I