On Sun, Jun 09, 2024 at 08:27:21PM +0800, Qian Yun wrote:
> radicalSolve(x^5-1,x) gives results with sin and cos,
> but (x^5-1)= (x-1)*(x^4+x^3+x^2+x+1), so the roots can be
> expressed with only radicals.
>
> (BTW, radicalSolve(x^4+x^3+x^2+x+1,x) also gives sin/cos.)
>
> I think for consistency, "radicalSolve" should not give
> answers with sin/cos (aka cyclotomic roots) when there's
> radical root available.
>
> This can be solved by tweaking "solveInner" in sovlerad.spad.
That is tricky question. It is well known that roots of 1 can
be expressed in terms of radicals. However, explict expression
usually is _much_ more complicated. So using roots of 1 is
a compromise, it is shorthand notation which we could in
principle convert to explicit (usually nested) radicals.
Also, roots of 1 are easy to recoginize when given in exponential
or trigonomotric form. Not so when expressed purely in terms
of radicals.
--
Waldek Hebisch
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