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. :-)
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Let be a 5-th root of unity then the other primitive roots of unity are: . Therefore, . This means if then and so . Now apply the argument above.
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