Irreducibility in finite field implies irreducibility in $\mathbb{Z}$? Let $p$ be a prime, $f$ be a polynomial of $\mathbb{Z}[x]$. Suppose that $f$ is irreducible in $\mathbb{F}_p[x]$. 
My question is : Is $f$ irreducible in $\mathbb{Z}[x]$ ? 
This question is originally came from the example 4.29 from the notes Field and Galois theory of J.S.Milne, in that he gave an implication of " a polynomial is irreducibile mod 3(still reducible mod 2) then it is irreducible in $\mathbb{Z}$.
Update I have more question after reading the comments and answer below. So, if $f$ is monic, what is the condition for irreducible mod $p$ for some prime $p$ implying irreducibility of $f$ over $\mathbb{Z}[x]$ ? Must it be "irreducible modulo $p$ for all the prime $p$ (not for some) ?

Certainly not. For example $x+px^2$ is irreducible mod $p$, as it is equivalent to $x$.
