Show a sum of 5th roots of unity is a real number.

Question

Define $\omega = e^{\frac{2\pi i}{5}}$, and let $u = -\omega^{2}(1+\omega)$ be a unit in $\mathbb{Z}[\omega]$. I need to show $u$ is a positive real number. I tried using the fact that $0=1+\omega +\omega^{2}+\omega^{3}+\omega^{4}$ so then $-\omega^{2}-\omega^{3}=u=1+\omega +\omega^{4}$. Is there a property for roots of unity to restrict them to being $\leq 1$?

Answer 1

Using the fact that $\omega^5 = 1$,

\begin{align*} u &= -\omega^2 - \omega^3 \\ &= - \omega^2 - \frac{1}{\omega^2} \end{align*}

Now try to relate this to $\omega$'s complex conjugate, using the fact that $\omega$ is unimodular. Spoilers ahead.

$$= -\left(\omega^2 + \overline{\omega^2}\right) = -2 \Re \omega^2 = -2 \cos\frac{4\pi}{5} > 0.$$