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 $d?$

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    $\begingroup$ I'm a little confused by your phrasing; is the question what is the maximum $n$ such that $\alpha^n \in \mathbb{R}$? You can certainly get $n$ such that $\varphi(n) \le d$. $\endgroup$ – Qiaochu Yuan Aug 13 '16 at 17:44
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    $\begingroup$ Adopting Qiaochu's interpretation of the question, it seems you can get arbitrarily large orders from $x^2+2x+b$ with a large $b$: the phase will be close to $\pi/2$, but just slightly off. (is it even clear that some power of $\alpha$ will always be real?). $\endgroup$ – Christian Remling Aug 13 '16 at 18:18
  • $\begingroup$ @QiaochuYuan The maximum minimum $n,$ yes.. $\endgroup$ – Igor Rivin Aug 13 '16 at 19:06
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    $\begingroup$ @IgorRivin: I don't understand your argument, but if you take $b=4$, say, then $-\alpha=1+i\sqrt{3}$ (this has phase $\pi/3$), so $\alpha^3\in\mathbb R$, $\alpha^2\notin\mathbb R$. $\endgroup$ – Christian Remling Aug 13 '16 at 19:15
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    $\begingroup$ For an explicit example, $3+4i$ is a root of $X^2-6X+25$ and has no real power (otherwise, $(3+4i)/5$ would be a root of unity, which is not the case). $\endgroup$ – YCor Aug 13 '16 at 22:42

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...


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