Lemma 8.1 (Hensel’s Lemma). Assuming that:

$R$ is complete with respect to an ideal $I$
$F \in R [X]$ , $s \geq 1$
$a \in R$ satisfies $F (a) \equiv 0 (m o d I^{s})$ , $F^{'} (a) \in R^{\times}$

Then there exists a unique

b \in R

such that

F (b) = 0

and

b \equiv 0 (m o d I^{s})

.