Usually Maple gives solutions in terms of radicals only up to degree=4
(for example using RootOf + allvalues).
Using 'irreduc' (to test first) and 'galois' (to check for the Galois
group, if degree <= 9) I have cases, where the result implies, that by
theory the roots can be given through radicals:
Though the results are difficult to read in my case they are 'C(6)' or
'C(8)', the cyclic groups of that order - thus abelian and IIRC those
groups are solvable.
NB: I did not find a way to ask Maple for 'isabelian( C(6) )' ... and
do not want to artificially write it as permutation group or so. And
I miss a command 'issolvable'.
Is there some extension package for Maple, which would provide that
explicit root representation (ok, may give results of useless size)?
32*c^5-16*c^4-32*c^3+12*c^2+6*c-1 <---> C(5)
64*c^6-32*c^5-80*c^4+32*c^3+24*c^2-6*c-1 <---> C(6)
Edited 02. Mar 2011: due to the discussion I should say, that the sextic
is for cos(Pi/13) and the quintic comes through cos(Pi/11).