[Math] polynomial with all roots on the unit circle

Question

I'm wondering if the following statement is true:
if all roots of a polynomial with integer coefficients are on the unit circle, then these roots are in fact roots of unity and the polynomial is a product of cyclotomic polynomial.

Answer 1

This isn't true. Take for example polynomial $5x^2-6x+5$. It's easy to check it has roots $\frac{3}{5}\pm\frac{4}{5}i$, which are both on the unit circle, but neither is a root of unity.

However, if you restrict your attention to monic integer polynomials, then this is indeed correct: it's a result due to Kronecker, and you can see a few proofs of this here.