What is the value of the sum \(\sum z\) where z ranges over all 7 solutions (real and nonreal) of the equation z^7 = -1?
Use roots of unity!
z^7 = 1 x (-1)^7
\(z_k = -1 \cdot e^{\frac{2k\pi}{n}i}\)
The sum of the roots of unity is 0 by vieta's formula. So -1(0) = 0.
