This looks like RR should be fixed to match AA here.

On 28 August 2024 12:40:49 BST, Kwankyu Lee <ekwan...@gmail.com> wrote:
>sage: ZZ(-1).nth_root(3)
>-1
>sage: _.parent()
>Integer Ring
>sage: QQ(-1).nth_root(3)
>-1
>sage: _.parent()
>Rational Field
>sage: RR(-1).nth_root(3)
>-1.00000000000000
>sage: _.parent()
>Real Field with 53 bits of precision
>sage: CC(-1).nth_root(3)
>0.500000000000000 + 0.866025403784439*I
>sage: _.parent()
>Complex Field with 53 bits of precision
>sage: QQbar(-1).nth_root(3)
>0.500000000000000? + 0.866025403784439?*I
>sage: _.parent()
>Algebraic Field 
>sage: AA(-1).nth_root(3)
>-1
>sage: _.parent()
>Algebraic Real Field
>
>sage: ZZ(-1)^(1/3)
>(-1)^(1/3)
>sage: QQ(-1)^(1/3)
>(-1)^(1/3)
>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: QQbar(ZZ(-1)^(1/3))
>0.500000000000000? + 0.866025403784439?*I
>sage: QQbar(QQ(-1)^(1/3))
>0.500000000000000? + 0.866025403784439?*I
>sage: AA(-1)^(1/3)
>-1
>
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