Polynomials with integer roots

Abstract

Let ${\mathcal{F}}_n$ be the set of unitary polynomials of degree $n\ge 2$ that have their roots in ${\mathbb{Z}}^{\ast }$. We note $$Q(x):=x^n+a_1{x}^{n-1}+\cdots +a_n.$$ We show that any two fixed consecutive coefficients $(a_j,a_{j+1})$ ($j\in \{1,\dots ,n-1\})$ define finitely many polynomials of ${\mathcal{F}}_n$.