TITLE: Polynomials with roots on, outside but not inside the unit circle
QUESTION [7 upvotes]: In relation to the (possible non-)existence of non-contracting ergodic endomorphisms on the torus, I have the following question on polynomials:  If the polynomial is monic, with integer coefficients and has no root strictly inside the unit circle, is it then also true that roots on the unit circle are roots of unity?
For polynomials up to degree 4 this seems to be the case, but I'd be happy with a higher degree counter-example. I am aware that Kronecker's Theorem says the answer is "yes" if all roots are on the unit circle.

REPLY [12 votes]: Let $f(z)$ be a degree-$n$ irreducible polynomial with integer coefficients and roots on the unit circle but no roots strictly within it. Then $g(z) := z^nf(1/z)$ is also a degree-$n$ irreducible polynomial, whose roots on the unit circle are the same as those of $f$. Since $f$ is the minimal polynomial of those unit roots and $g$ has the same degree as $f$, $g$ must also have the same roots as $f$.
Now since $f$ has no roots within the unit circle, $g$ has by construction no roots outside the unit circle, so all roots of $f$ lie on the unit circle. The rest follows from Kronecker's theorem.