The sum of cube roots of unity (1+w+w^2) is

Let . By DeMoivre's theorem, we have



where is any integer, but we can capture all the cube roots of unity by selecting , and in particular when we get 1.




Recall that , so that . On the other hand, , so . So in , we find the imaginary parts cancel, leaving us with



for which we have



so the sum is .

Another way of computing the sum is to notice that



for any . We assumed that , so the RHS is 0. Then for , we must have .