Hensel's lemma

Builds on Congruence modulo n Prime number

Statement

Let $p$ be a prime number, 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.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Divisibility definition
Congruence modulo n definition
Prime number definition
Hensel's lemma statement

Tag	`010QVB71T2`
URI	`m2://010QVB71T2`
Kind	statement, theorem
Writing	W1 stated
MSC	11A07
Status	This copy of the site has no build data. Run `m2 status 010QVB71T2`.
Uses	Congruence modulo n, Prime number
Same as	Wikidata Q1424496 Wikipedia Hensel's lemma

Hensel's lemma

Statement

Remarks

History

Reading path

Further reading

What this page needs for W2

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