Irreducibility Implication between $\mathbb{Q}[x]$, $\mathbb{Z}[x]$, and $\mathbb{Z_p}[x]$

Question

These following questions might be pretty easy, but I am really confused about them.

Let $f(x)$ $\in \mathbb{Z}[x]$, $p=$prime.

1- Does irreducibility of $f(x)$ in $\mathbb{Z_p}[x]$ imply irreducibility of $f(x)$ in $\mathbb{Q}[x]$ ?

2- Does irreducibility of $f(x)$ in $Z[x]$ imply irreducibility of $f(x)$ in $\mathbb{Z_p}[x]$?

3- Does irreducibility of $f(x)$ in $\mathbb{Q}[x]$ imply irreducibility of $f(x)$ in $\mathbb{Z_p}[x]$ ?

Answer 1

It is a theorem that $f(x) \in \mathbb Z[x]$ reducible over $\mathbb Q$ implies $f$ is reducible over $\mathbb Z$.

As a consequence, if $f$ is not reducible over $\mathbb Z$ it is also not reducible over $\mathbb Q$.

As for the other cases: do you assume $\operatorname{deg}{(f)} \ge 1$?

And what assumptions do you make on the degree of $\overline{f}$ (that is, $f$ with coefficients reduced $\mod p$)?

Answer 2

If $f(x)$ is irreducible in $\mathbb{Z_p}[x]$, it does not imply that $f(x)$ is irreducible in $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$. For example, let $f(x)= px^{2}+x$, then: $f(x)= x$ in $\mathbb{Z_p}[x]$ and so is irreducible in $\mathbb{Z_p}[x]$, while $f(x)= px^{2}+x=x(px+1)$ in $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$, and so is reducible in $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$. But if $f(x)$ is monic and irreducible in $\mathbb{Z_p}[x]$, then it is irreducible in $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$.