I am going in unit circles (get the pun) with this problem.
But I managed to make this problem more complicated than it should be. Instead of working with cyclotonomic polynomials, as I should have. I reduced the problem to trigonometry which makes it more difficult.
I am sure you know what I am about to say but let me just post it.
A cyclotonomic polynomial is,
\Phi_n(z) = 1+z+z^2+...+z^n
We can factor them as,
(z-z_1)(z-z_2)+...+(z-_n)
Where,
z_1,z_2,...,z_n
Are points on the regular (n+1)-gon inscribed in the unit circle in the complex plane (except unity).
Basically, I am saying this is the cyclotonomic polynomial of degree n evaluated at 1.