Existence of polynomials in $\mathbb{Q}[x]$ and $\mathbb{Z}[x]$ with same splitting fields.

Question

I've been asked to prove the following statement in a Galois Theory Seminar after being introduced to Dedekind's Theorem. (I assume this could potentially help getting the answer.)

Let $f(x)$ be a a monic polynomial of degree $N$ in $\mathbb{Q}[x]$ and let $E_f$ be its splitting field over $\mathbb{Q}$. Then, there exists a monic polynomial $p(x) \in \mathbb{Z}[x]$ of degree $N$ that has the same splitting field $E_f$.

I can't find any way to prove it.

Any help is welcome!

EDIT: There used to be what I though was a counter example that has been answered already.

Answer 1

Let $x^n + a_1 x^{n_1}+ \cdots + a_n=0$, and write $y=Nx$; then $y^n + a_1 N y^{n_1}+ \cdots + a_n N^n =0$. Write $a_j=p_j/q_j$ and $N=\prod_j q_j$ and you are done. This is basically Gauss's lemma isn't it?