Sum of the $n$-th roots of unity

Geometric series. Since $\omega\neq1$ (for $n\ge2$) and $\omega^n=1$:
$$\sum_{k=0}^{n-1}\omega^k=\frac{1-\omega^n}{1-\omega}=0.$$

Conclusion. The sum of all $n$-th roots of unity is $0$.

Geometric interpretation. The centroid of a regular $n$-gon inscribed in the unit circle is the origin.