Is it immediately apparent that the solution to the system of equations,
can be given by the roots of unity? Specifically,
where the are,
Example. Let . Then is equivalent to,
Expanded out, is a -deg polynomial and its 16 roots are given by where,
Ramanujan considered the system for in the general case and also as nested radicals. For , see this related post. (Interestingly, in the general case is no longer completely solvable in radicals.)
I observed empirically. How do we prove from first principles that this is indeed the solution?