Find polynomial given splitting field

Question

Let $f\in\mathbb{Q}[x]$ a monic polynomial such that $f$ has degree $n$. Let $E_f$ be the splitting field of $f$ over $\mathbb{Q}$. I would like to show that there exists a monic polynomial in $\mathbb{Z}[x]$ of degree $n$ such that it has the same splitting field. I don't even know how to tackle this problem. Any help would be appreciated. Edit: as has been pointed out, it would suffice to prove that $f(x)$ and $q^n=f(x/q)$ have the same splitting field for any integer $q$. This is clear since the roots of $f$ in $E_f$ are the same than those pf $f(x/q)$ except for multiplying by a rational constant. Am I right?