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Old 2020-04-13, 20:29   #9
Nick
 
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Dec 2012
The Netherlands

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OK, I think I understand the confusion.
Let's use your example of q=7 with \(\zeta=e^{2\pi i/7}\).
Then we have
\[ \zeta^0=1,\zeta^1=\zeta,\zeta^2,\zeta^3,\zeta^4,\zeta^5,\zeta^6\]
all distinct and \(\zeta^7=1\) again.
We also have the equation
\[ \zeta^6+\zeta^5+\zeta^4+\zeta^3+\zeta^2+\zeta+1=0\]
so it follows that
\[ -1=\zeta^6+\zeta^5+\zeta^4+\zeta^3+\zeta^2+\zeta \]
Thus you can use \(\zeta^0\) to \(\zeta^5\) inclusive OR \(\zeta^1\) to \(\zeta^6\) inclusive and still equate coefficients (they are linearly independent).

I hope this helps!
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