Hi everyone,
How can I prove that Q(Zeta36) = Q(Zeta12,Zeta18) does not contain ANY primitive 5th roots of unity?
[here, ZetaN = cos2pi/N + isin2pi/N ]
Many many thanks in advance. x
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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