It is well known that if $x$ is a rational multiple of $\pi$ then $\cos x$, $\sin x$, etc, are algebraic numbers. What is known about the inverse problem?
That is, is there a set of conditions that if imposed on $\alpha$ imply that $\cos^{-1} \alpha$ is a rational multiple of $\pi$?
Another way of putting it is, given the complex number $z=a+ib$, is there a way of deciding whether there exists some $n$ such that $z^n=1$?
EDIT: 'factor' -> 'multiple'