Question

Find the seven seventh roots of unity and prove that the sum of their nth powers always vanishes unless n be a multiple of seven, n being an integer and then the sum is seven.

Answer 1

Already proved in B (c), P. 60.
As in Q.6, the m, $n^{th}$ roots or unity are
1, $\alpha$, ${\alpha }^2$, ....., ${\alpha }^{n-1}$ where
$\alpha$ = cos(2$\pi$/n) + i sin (2$\pi$/n).
We have to find the sum of the $p^{th}$ powers of these roots.
$\therefore 1^p+{\alpha }^p+({\alpha }^2{)}^p+({\alpha }^3{)}^p+.....({\alpha }^{n-1}{)}^p$
$=1^p+{\alpha }^p+{\alpha }^{2p}+{\alpha }^{3p}+.....{\alpha }^{(n-1)p}$
$=\frac{1-({\alpha }^p{)}^n}{1-{\alpha }^p}$
[Summing the G.P. of common ratio ${\alpha }^p$]
$=\frac{1-{\alpha }^{pn}}{1-{\alpha }^p}$
$=\frac{1-\left[cos\right(2\pi /n)+isin(2\pi /n){]}^{pn}}{1-\left[cos\right(2\pi /n)+isin(2\pi /n){]}^p}$
$=\frac{1-cos2\pi p-isin2\pi p}{1-cos\left(2\pi /n\right)-isin\left(2\pi p/n\right)}$
$=\frac{1-1-i.0}{1-cos\left(2\pi p/n\right)-isin\left(2\pi p/n\right)}$
$=\frac{0}{1-cos\left(2\pi p/n\right)-isin\left(2\pi p/n\right)}=0$
If p is not a multiple of n and hence $D^r\ne 0$
If p is a multiple of n, say p = nk
${\alpha }^{mp}=,\left[cos\right(2\pi /n)+isin(2\pi /n){]}^{mnk}$
$=cos\frac{2\pi mnk}{n}+isin\frac{2\pi mnk}{n},$
$1\le m\le n-1$
$=cos2\pi mnk+isin2\pi mnk=1+io=1.$
So in this case, each term of the series in (1) is 1.
Hence the sum of the series (1) is n.