Roots of Unity with Rational Real Parts All of the $4^{\text{th}}$ and $6^{\text{th}}$ roots of unity have real parts that are rational numbers. Are these the only roots of unity $z$ such that $\text{Re}(z)\in \mathbb{Q}$  ?

Suppose $z^n=1$ and $\Re(z)\in \mathbb{Q}$. Then also $\left(\overline{z}\right)^n=1$, so $z$ and $\overline{z}$ are both algebraic integers. This means that $z+ \overline{z} = 2 \Re(z)$ is also an algebraic integer. The only algebraic integers that are rational are integers, so $2 \Re(z)$ must be an integer. Since $z$ lies on the unit circle we find that the only possible values for $2\Re(z)$ are $\{-2,1,0,1,2\}$. These values indeed correspond to the 6th roots of unity together with $\pm i$.
