roots of unity – The Accidental Mathematician

A root of unity is a complex number $z$ such that $z^n=1$ for some positive integer $n$ . This means that $z=e^{2\pi i k/n}$ for some integer $k$ ; if $k$ is relatively prime to $n$ , we say that $z$ is a primitive $n$ -th root of unity, meaning that $z$ is not a $m$ -th root of unity for any $1\leq m<n$ .

Here’s a question: when can we have

$(1) \ \ z_1+\dots +z_\ell=0$

if $z_1,\dots,z_\ell$ are roots of unity?

This is a little bit vague, in that I did not say what kind of tentative characterization we are looking for. If you were inclined to play devil’s advocate, you could say that equation (1) provides a good enough description. There are, however, less obvious answers that have come in handy in various parts of my research, so let’s look at some of them.

Tag: roots of unity

Vanishing sums of roots of unity