Change behavior of algebraic reals with respect to fractional powers

[kwankyu]

We fix two inconsistent behaviors of algebraic reals:

(1)

sage: RR(-1)^(1/3) 
0.500000000000000 + 0.866025403784439*I
sage: CC(-1)^(1/3)
0.500000000000000 + 0.866025403784439*I
sage: QQbar(-1)^(1/3)
0.500000000000000? + 0.866025403784439?*I
sage: AA(-1)^(1/3)                   
-1

See the discussion: https://groups.google.com/g/sage-devel/c/s81ieq2vpVo

(2)

sage: QQbar((-1)^(1/3))
0.500000000000000? + 0.866025403784439?*I
sage: RR((-1)^(1/3))
...
TypeError: unable to convert '0.500000000000000+0.866025403784439*I' to a real number
sage: AA((-1)^(1/3))                
-1

With this PR,

sage: AA(-1)^(1/3)  # principal cube root
0.500000000000000? + 0.866025403784439?*I
sage: AA((-1)^(1/3))
...
ValueError: Cannot coerce algebraic number with nonzero imaginary part to algebraic real

These behavior changes fix #12745, #36735.

On the other hand, the behavior of AA(x).nth_root(n) does not change, and gives an n-th root in AA.

sage: AA(-8).nth_root(3)
-2
sage: AA(8).nth_root(3)
2
sage: AA(sqrt(8)).nth_root(3)
1.414213562373095?
sage: AA(-sqrt(8)).nth_root(3)
-1.414213562373095?

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

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[dimpase]

As I explained in sage-devel, I don't think the behaviour of AA(-1)^(1/3) is incorrect.
Indeed, $1/(2k+1):AA\to AA$ is 1-1 on the whole AA (or, indeed, on the reals).

[kwankyu]

The issue is not mathematical correctness. Any cube root of -1 is a mathematically correct answer to AA(-1)^(1/3).

Currently, for a parent P, P(-1)^(1/3) gives the principal root of -1 for all parents P, with a unique exception AA. This inconsistency causes confusion. Fortunately, P(x).nth_root(3) may be used when a cube root in P is expected.

In the same vein, I think it is a "bug" if P(-1).nth_root(n) gives an answer not in P. Currently RR(-1).nth_root(4) correctly raises an exception. But AA(-1).nth_root(4) silently gives the principal root of -1. One may hope to "fix" this. This PR won't.

If you are still not convinced, please put forth your objection in #38362 when that PR is ready to be merged.