Factoring Monic Polynomials in $\mathbb{Z}[x]$ over $\mathbb{Z}_p$ Suppose that $f \in \mathbb{Z}[x]$ is irreducible. Then $f$ factors as $\prod\limits_{i=1}^n {f_i}^{e_i}$ over $\mathbb{Z}/p \mathbb{Z}$ and by Hensel's Lemma, each term ${f_i}^{e_i}$ lifts to some factor $F_i$ of $f$ over $\mathbb{Z}_p$.
Must $F_i$ be irreducible over $\mathbb{Z}_p$? 

No. For example consider the case $f(x)=x^2+7$, $p=2$. Modulo $p$ we have
$$
f(x)\equiv x^2+1\equiv(x+1)^2.
$$
The lifted factor in $\mathbb{Z}_2[x]$ has to be $f(x)$ itself. But there is a $w=\sqrt{-7}$ in the ring $\mathbb{Z}_2$, so the lifted polynomial factors as $(x-w)(x+w)$ over the 2-adics.
