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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?

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    $\begingroup$ Show that for any integer $q$ the splitting fields of $f(x)$ and $q^nf(x/q)$ are the same. With a suitable choice of $q$ the latter is in $\Bbb{Z}[x]$. $\endgroup$ – Jyrki Lahtonen Nov 18 '18 at 20:09
  • $\begingroup$ But it would not be monic $\endgroup$ – Ray Bern Nov 18 '18 at 20:11
  • $\begingroup$ Thank you. You're right $\endgroup$ – Ray Bern Nov 18 '18 at 20:16

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