radical roots and cyclotomic roots

[Qian Yun]

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.

- Qian

[Waldek Hebisch]

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

[Qian Yun]

But for consistency, compare:

radicalSolve(eval(x^4+x^3+x^2+x+1,x=y),y)

radicalSolve(eval(x^4+x^3+x^2+x+1,x=y+1),y) -- or x=2*y


Also, with your commit fcacacff "Improve 'radicalSolve'",
the roots for degree 4 cyclotomic polynomial is no longer
monstrous.

I'm just talking about x^5-1 here, involving roots of
degree 4 polynomials. Not talking about x^17-1.

- Qian