---
tag: 010QVB71T2
kind: statement
role: theorem
name: Hensel's lemma
msc: [11A07]
uses: [0TPW25CBN6, 0SN0MAEKYX]
sources:
  - url: https://en.wikipedia.org/wiki/Hensel%27s_lemma
    locator: Lifting simple roots
same_as:
  wikidata: Q1424496
  wikipedia: Hensel's lemma
---

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]], and let $f$ be a polynomial with integer coefficients, with derivative $f'$.
Let $r$ be an integer such that $f(r) \equiv 0 \pmod p$ and $f'(r) \not\equiv 0 \pmod p$.
Then for each positive integer $k$ there is an integer $r_k$ with $r_k \equiv r \pmod p$ and $f(r_k) \equiv 0 \pmod{p^k}$.
This $r_k$ is unique modulo $p^k$.

## Remarks

The proof gives the lifts step by step. If $f(r_k) \equiv 0 \pmod{p^k}$ and $t$ is an inverse of $f'(r)$ modulo $p$, then $r_{k+1} = r_k - t f(r_k)$ is a root modulo $p^{k+1}$.
For example, $3^2 \equiv 2 \pmod 7$ and $f'(3) = 6$ for $f = x^2 - 2$. The step gives $3 - 6 \cdot 7 = -39 \equiv 10 \pmod{49}$, and $10^2 = 100 \equiv 2 \pmod{49}$.
In the language of $p$-adic numbers, a simple root modulo $p$ lifts to a unique root in the $p$-adic integers.
The lemma is the analogue of Newton's method for congruences.

## History

The lemma is named after Kurt Hensel.

## Explanations

### L1 human

A solution of a polynomial congruence modulo $p$ often gives solutions modulo $p^2$, $p^3$, and so on.
The condition $f'(r) \not\equiv 0 \pmod p$ says that $r$ is a simple root modulo $p$.
Under this condition, each step of the lifting has exactly one answer.