Theorem 4.4 (Hensel’s Lemma version 2). Assuming that:

$(K, | ∙ |)$ is a complete discretely valued field
$f (X) \in O_{K} [X]$
$\bar{f} (X) : = f (X) (m o d m) \in k [X]$ factorises as $\bar{f} (X) = \bar{g} (X) \bar{h} (X)$ in $k [X]$
$\bar{g} (X)$ and $\bar{h} (X)$ coprime.

Then there is a factorisation

f (X) = g (X) h (X)

in $O_{K} [X]$ , with $\bar{g} (X) \equiv g (X) (m o d m)$ , $\bar{h} (X) \equiv h (X) (m o d m)$ and $\deg \bar{g} = \deg g$ .