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 $d?$