Problem:
Given a polynomial $f(x)=c_0+c_1\,x+\ldots+c_n\,x^n\in\mathbb{Z}[x]$, assume that exists $p\in\mathbb{Z}$ prime that satisfies:
- $p\,\nmid\,c_n$
- $p\,\mid\,c_i,\;\forall i=0,\,\ldots,n-1$.
- $p^2\,\nmid\,c_k\; $ for at least one index $k$, $\; 0\leq k\leq n-1$.
Take the minimum $k$ that satisfies (3) and denote it by $k_0$. Suppose that there is a factorization of $f(x)$ in $\mathbb{Z}[x]$:
$f(x) = g(x)\,h(x).$
Show that
$\min\left\{\deg(g(x)), \deg(h(x))\right\}\leq k_0$
Observation
If I consider the projection morphism: $\phi:\,\mathbb{Z}[x] \longrightarrow \mathbb{Z}/p^2[x]$, that sends the coefficients of a polynomial to their class, I see that this morphism preserves the degree of the polynomials $f,\,g$ and $h$, because the class of their leading terms is non-zero. For $\phi(f(x))$, all terms of index less than $k_0$ are zero. I'm looking for applying Eisenstein here and find a contradiction. However, $\mathbb{Z}/p^2$ is not a UFD...
Question
I just want some clue. Because I don't know where to start.