Exercise 17, § 11

Let $P_n$ be the set of monic polynomials $X^n + a_1 X^{n-1} + \ldots + a_n$, of $\mathbf{Z}[X]$ whose roots in $\mathbf{C}$ all have absolute value $\leq 1$. a) Show that the set $P_n$ is finite. Deduce that for every polynomial $F \in P_n$, if $F_h$ denotes the polynomial whose roots are the h-th powers of the roots of $F$, then there exist two distinct integers $h, k$ such that $F_h = F_k$. b) Conclude from a) that all the roots of a polynomial $F \in P_n$ are either zero or roots of unity (* Kronecker's theorem »).