When is a sum of consecutive roots of unity an integer

Question

Let $\xi \neq 1$ be an $n$th root of unity. When is a sum of the form $$ 1+\xi+\xi^2+\ldots+\xi^r, \quad 1 \leq r \leq n-1, $$ an integer? What are the possible integers?

I suspect that the answers to both questions are the trivial ones: $r=n-1$ (or $\xi$ is not primitive and $r \mid (n-1)$) and the sum is $0$ in all cases.

Answer 1

The following may be helpful for you:

$$ 1+\xi+\xi^2+\ldots+\xi^r=\frac{1-\xi^{r+1}}{1-\xi}$$

You can discuss when $\frac{1-\xi^{r+1}}{1-\xi}$ is an integer, ecpeacially $\xi$ is an irrational number.

Answer 2

Suppose $n\ne0$ or 1. Then $\xi^{r+1}-1=n(\xi-1)$. Now look at the geometry. Let $K$ be the point on the unit circle (in the Argand diagram) such that $\xi-1$ and $\xi^{r+1}-1$ lie on the ray $OK$. We must have the three distinct points 1, $\xi,\xi^{r+1}$ of the circle collinear. Contradiction.

Doing this on an iPhone6+ is challenging. Is that right?