On 28 August 2024 12:28:54 BST, Kwankyu Lee <ekwan...@gmail.com> wrote:
>
>
>On Wednesday, August 28, 2024 at 7:57:43 PM UTC+9 john.c...@gmail.com wrote:
>
>Surely the output of -1 for AA(-1)^(1/3) is correct: AA is the "Algebraic
>Real Field" and -1 has exactly one cube root in there, namely itself. On
>the other hand, QQbar(-1) has 3 cube roots and one is chosen (in some
>deterministic way).
>
>I do not think that AA(-1)^(1/3) should return a cubroot in another
>field/parent when there is one in the same field/parent. Compare
>
>sage: QQ(-1).nth_root(3)
>-1
>sage: RR(-1).nth_root(3)
>-1.00000000000000
>sage: CC(-1).nth_root(3)
>0.500000000000000 + 0.866025403784439*I
>
>which is as it should be (in my opinion!)
>
>
>x^(1/n) and x.nth_root(n) do not behave in the same way. All x.nth_root(n)
>gives an n-th root in the same field to which x belongs while all x^(1/n)
>gives the primitive n-th root of unity, with the exception of AA(-1)^(1/n).
the function 1/n : AA->AA is 1-1 on all AA, and I don't see why this property
should be broken.
Mathematically it makes perfect sense as is.
In fact, it should also work like this for any real field, not only AA.
Dima
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