I have this theorem in my algebra book but I cannot seem to come up nor find a proof that is not done by contradiction. Can you please help me? Maybe give me a hint ? I would really like to have a clue on how to start the proof by assuming that our polynomial is, in fact, irreducible over $\Bbb Z_p$ .
I found the same question and I will leave the link here: Mod $p$ irreducibility test proof question
Let $p$ be a prime and suppose that $f(x) \in \Bbb Z[x]$ with deg $f(x) \ge 1$. Let $f'(x)$ be the polynomial in $\Bbb Z_p[x]$ obtained from $f(x)$ by reducing all the coefficients of $f(x)$ modulo $p$. If $f'(x)$ is irreducible over $\Bbb Z_p$ and deg $f (x) =$ deg $f'(x)$, then $f(x)$ is irreducible over $\Bbb Q$.