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Is it true that an integer polynomial with constant term $\pm 1$ has at least one zero within the unit disc $|z|\leq 1$?

I am trying to solve a competitive math problem, and this fact being true would solve it. However, for some reason, it seems too good to be true.

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It looks pretty straightforward to me: by Viète's theorem, the product of the roots is $\pm\frac{1}{n}$ with $n\in\mathbb{Z}\setminus\{0\}$ being the leading coefficient. If all the roots were outside the unit circle, the product of the roots would have a modulus $>1$, contradiction.

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  • $\begingroup$ Yes you're right. Gosh. Should have thought of that. And that solves the problem I was trying to solve. $\endgroup$ – fierydemon Mar 9 '17 at 1:29

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