Galois groups of a a seperable polynomial in a splitting field over $\mathbb{Q}$ and $\mathbb{Z}_p$

Question

So here is my question. If a polynomial $f(x)\in\mathbb{Q}[x]$ is separable of degree $n$ and has Galois group $G\subseteq S_n$.Is it true that if $\bar{f}\in\mathbb{Z}_p[x]$ is separable then $\bar{G}\simeq G\subseteq S_n$. It seems true, but I am not sure. Any help is appreciated.

Answer 1

No. For instance, $f(x)=x^2+1$ has Galois group $\mathbb{Z}/2\mathbb{Z}$ over $\mathbb{Q}$, but $\bar{f}(x)=(x+2)(x+3)$ in $\mathbb{F}_5[x]$, hence its Galois group over $\mathbb{F}_5$ is trivial.

Moreover, since the Galois group of any finite extension of a finite field is cyclic, you don't have much hope in general that the Galois groups of $f$ and $\bar{f}$ will be isomorphic.