TITLE: order of reality
QUESTION [6 upvotes]: Suppose I have a monic polynomial $p(x) \in \mathbb{Z}[x],$ of degree $d,$ and $\alpha$ a nonzero root of $p(x).$ The question is: assuming that $\alpha$ has some real power, what is the maximum order of $\alpha$ in $\mathbb{C}^\times/\mathbb{R}^\times?,$ in terms of $d?$ Obviously, order $d$ is possible and achieved by $p(x) = x^d-1.$
EDIT since there appears to be some confusion, here is a restatement: suppose $\alpha$ is an algebraic integer of degree $d,$ with $\arg \alpha = \frac{2 \pi k}{l},$ with $k, l \in \mathbb{Z}$ in lowest terms. Can we bound $l$ in terms of $d?$

REPLY [8 votes]: The trivial bound is slightly above $d^2$. Note that $\beta=\alpha/\bar\alpha$ is a root of unity of pretty much the same order (give or take a factor of $2$) whose minimal polynomial has degree at most $d^2$. However, with the roots of unity, we know that the order $n$ corresponds to the degree $\varphi(n)$ (cyclotomic polynomials and such) and $\varphi(n)$ cannot be much less than $n$ on the rough power scale (but can be as many times smaller than $n$ as you wish, whence the discrepancy between $2$ and $3$ in Christian's example). I have no idea if the square is actually needed though...