Cyclotomic quotients of two conjugates of an algebraic number

Let $\alpha $ be an algebraic number of degree $d \geq 2$. We consider the set $E(\alpha)$ of positive integers $n$ such that the primitive $n$th root of unity $e^{2\pi i/n}$ is expressible as a quotient of two conjugates of $\alpha $ over ${\Bbb Q}$. In particular, our results imply that $E(\alpha )$ is small. We prove that $|E(\alpha )| < d^{\frac{c}{\log \log d}}$, where $c=1.04$ for each sufficiently large $d$. We also show that, in terms of $d$, this estimate is best possible up to a constant, since the constant $1.04$ cannot be replaced by any number smaller than $0.69$.