Roots of unity

The $n$th roots of unity are the complex numbers $z$ satisfying $z^n=1$. They are $1,ho,h{o}^2,\dots ,{\rho }^{n-1}$, where $\rho =e^{2\pi i/n}$. Their sum is $0$ whenever $n>1$, because $(1-\rho )(1+\rho +\cdots +{\rho }^{n-1})=1-{\rho }^n=0$ and $1-\rho \ne 0$. Their product is $(-1{)}^{n-1}$, since they are the roots of $z^n-1=0$ and the product of roots equals $(-1{)}^n$ times the constant term divided by the leading coefficient. To solve $z^n=a$, write $a=r{e}^{i\theta }$; then $z_k=\sqrt[n]{r}{e}^{i(\theta +2\pi k)/n}$ for $k=0,1,\dots ,n-1$.