Does irreducibility in $\mathbb{F}_p[x]$ imply irreducibility in $\mathbb{Q}[x]$?

Question

As per the title. I'm wondering if $f(x) \in \mathbb{Z}[x]$ is monic and has degree $\geq 2$ and $p$ is prime. I want to prove that if $f \mod n$ in $\mathbb{F}_p$ is irreducible, then $f$ is irreducible in $\mathbb{Q}[x]$. One approach I considered is that since irreducibility over rationals implies irreducibility over ring of integers. So, if I can show that if it's reducible in the integers, it is in $\mathbb{F}_p$ would be an equivalent statement. Could someone show me how I could prove this?

Answer 1

Suppose $\;f(x)\in\Bbb Z[x]\;$ reducible in $\;\Bbb Q[x]\;$ . Then by Gauss Lemma it is reducible over $\;\Bbb Z[x]\;$, say $\;f(x)=g(x)h(x)\;,\;\;g,h\in\Bbb Z[x]\;$ , but then

$$\;f(x)\pmod p=\left(g(x)\pmod p\right)\left( h(x)\pmod p\right)\implies f(x)\pmod p\in\Bbb F_p[x]\,$$

is reducible.