Problem: Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial whose roots are all real and lie in the interval $[-2,2]$. Prove that each root of $P$ has the form $2\cos (2\pi r)$ for some $r\in \mathbb{Q}.$
I think the following theorem of Kronecker is useful: If $\alpha$ is an algebraic integer such that $\alpha$ and all its conjugate have modulus at most $1$, then either $\alpha =0$ or $\alpha$ is a root of unity.
I have read some book that Kronecker solved the above problem by considering the map $z \mapsto z+z^{-1}$. I tried to Define the map $f$ from $\mu_n$, the set of n-th roots of unity, to the set $S$ of conjugate algebraic integers in $[-2,2]$ and prove it is a bijection. Then $f(\zeta_n^k)=2\cos (2\pi k/n) \in S$, but then I dont know how to complete the other direction, please helps. I think there is also a connection with the maximal real subextension $\mathbb{Q}(\zeta_n+\zeta_n^{-1})$.