# Find polynomial given splitting field

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?

• 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 $\Bbb{Z}[x]$. – Jyrki Lahtonen Nov 18 '18 at 20:09
• But it would not be monic – Ray Bern Nov 18 '18 at 20:11
• Thank you. You're right – Ray Bern Nov 18 '18 at 20:16