correct me if I am wrong, both the constituents of alpha and omega are conjugates right. So can I just use (z^2 -2Re(z)z + 1) and represent it as quadratics that way?
Completely correct, I know you got this the next part is knowing what to replace with think of it like this a cycle is and you want to divide by 5 to find all the roots of unity. Then apply the conjugate root theorem.