If $f(x)$ is irreducible in $ \mathbb z [ x]$ , then for all primes $p$ the reduction $f'(x)$ of $f(x)$ modulo $p$ is irreducible in $F_p[x]$.

Question

If $f(x)$ is irreducible in $ \mathbb z [ x]$ , then for all primes $p$ the reduction $f'(x)$ of $f(x)$ modulo $p$ is irreducible in $F_p[x]$.

Is the statement true? Can anyone give me a hint?

My attempt : I think the statement is not true.

$x^2+4x+5$ is irreducible in $\mathbb z[x]$ But so is not in $F_2[x]$.

Am I right? Can anyone please tell me on what more conditions the statement will be true?

Answer 1

This is false: for instance $1+X^2$ is irreducible in $\mathbb{Z}[X]$ but not in $\mathbb{F}_2[X]$, because there $1+X^2=(1+X)^2$.