Is there a way to describe/characterize all complex numbers $z$ with $|z| = 1$ such that $z$ is the root of some polynomial with non-negative integer coefficients?
For instance, I've found that such $z$ must (clearly) be algebraic, hence if $z = e^{i\theta}$ then $\theta$ must not be algebraic lest $z$ be transcendental (because $e^a$ is transcendental for any algebraic $a$).
But this is not a sufficient condition as there are algebraic numbers that aren't roots of polynomials with non-negative integer coefficients, as proven in this SE post.