User:Barto/Hensel's Lemma/First Form

Theorem
Let $p$ be a prime number.

Let $k>0$ be a positive integer.

Let $f(X) \in \Z[X]$ be a polynomial.

Let $x_k\in\Z$ such that:
 * $f(x_k)\equiv 0 \pmod{p^k}$.
 * $f'(x_k)\not\equiv 0 \pmod{p}$.

Then for every positive integer $l>0$ there exists an integer $x_{k+l}$ such that:
 * $x_{k+l}\equiv x_k\pmod{p^k}$
 * $f(x_{k+l})\equiv 0 \pmod{p^{k+l}}$

Moreover, each such $x_{k+l}$ is unique up to a multiple of $p^{k+l}$.