# Hensel's Lemma

## Lemma

### First Form

Let $p$ be a prime number.

Let $k > 0$ be a positive integer.

Let $f \left({X}\right) \in \Z \left[{X}\right]$ be a polynomial.

Let $x_k \in \Z$ such that:

$f \left({x_k}\right) \equiv 0 \pmod {p^k}$
$f' \left({x_k}\right) \not \equiv 0 \pmod p$

Then for every integer $l \ge 0$, there exists an integer $x_{k + l}$ such that:

$f \left({x_{k + l} }\right) \equiv 0 \pmod {p^{k + l} }$
$x_{k + l}\equiv x_k \pmod {p^k}$

and any two integers satisfying these congruences are congruent modulo $p^{k + l}$.

Moreover, for all $l\geq0$ and any solutions $x_{k + l}$ and $x_{k + l + 1}$:

$x_{k + l + 1} \equiv x_{k + l} - \dfrac {f \left({x_{k + l} }\right)} {f' \left({x_{k + l} }\right)} \pmod {p^{k + l + 1} }$
$x_{k + l + 1} \equiv x_{k + l} \pmod {p^{k + l} }$

### Composite Numbers

Let $b \in \Z \setminus \set {-1, 0, 1}$ be an integer.

Let $k > 0$ be a positive integer.

Let $\map f X \in \Z \sqbrk X$ be a polynomial.

Let $x_k \in \Z$ such that:

$\map f {x_k} \equiv 0 \pmod {b^k}$
$\gcd \set {\map {f'} {x_k}, b} = 1$

Then for every integer $l \ge 0$, there exists an integer $x_{k + l}$ such that:

$\map f {x_{k + l} } \equiv 0 \pmod {b^{k + l} }$
$x_{k + l} \equiv x_k \pmod {b^k}$

and any two integers satisfying these congruences are congruent modulo $b^{k + l}$.

Moreover, for all $l \ge 0$ and any solutions $x_{k + l}$ and $x_{k + l + 1}$:

$x_{k + l + 1} \equiv x_{k + l} - \dfrac {\map f {x_{k + l} } } {\map {f'} {x_{k + l} } } \pmod {b^{k + l + 1} }$
$x_{k + l + 1} \equiv x_{k + l} \pmod {b^{k + l} }$

## Source of Name

This entry was named for Kurt Wilhelm Sebastian Hensel.