One minor point though - you have done the question for Zeta_5 = cos2pi/5+isin2pi/5.
But the original question was for Zeta = ANY PRIMITIVE 5th root of unity.
So my question is does Q(Zeta) = Q(Zeta_5) ? Because if so then your argument still holds. :-)
But I am all alone, , no girl wants to spend time with me.
This is Mine 12,5th Post!!!