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Arrange these points on the complex unit circle centered at 0 with W = W_0 = 1. The resulting complex values {W_i} of the N points are known as the Nth roots of unity, as for each i=0,1,...,N-1, we have (W_i)^N = 1. In other words, these are the N roots of the polynomial z^N-1.
Given the roots (and multiplicities) for a polynomial, we can factor it. In this case,
z^N-1=(z-1)(z-W_1)(Z-W_2)...(z-W_(N-1)).
We also remember the following identity from algebra:
(z^N-1)/(z-1)=z^(N-1)+z^(N-2)+...+z^2+z+1.
Putting these together gives
(z-W_1)(z-W_2)...(z-W_(N-1))=z^(N-1)+z^(N-2)+...+z^2+z+1.
If we let z=1, the left side of this identity becomes precisely the product of the diagonals, while the right side becomes the sum of N 1's. Thus, our answer is N. |
