Hensel's lemma

亨泽尔引理

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Translation evidence: Literature usage

Meaning in context

For prime $p$ and $f\in\mathbb Z_p[X]$, a simple root $a$ of $f\bmod p$ lifts to a unique root $\alpha\in\mathbb Z_p$ with $\alpha\equiv a\pmod p$.

对素数 $p$ 及 $f\in\mathbb Z_p[X]$，$f$ 模 $p$ 的单根 $a$ 可唯一提升为 $\mathbb Z_p$ 中满足 $\alpha\equiv a\pmod p$ 的根。

Example

The simple root $3$ of $X^2-2$ modulo $7$ lifts to a $7$-adic root congruent to $10$ modulo $49$.

Usage and distinctions

The derivative must be nonzero modulo p in this version; multiple roots require stronger hypotheses or different lifting statements.

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Definition sources

1. Keith Conrad — Hensel's Lemma, University of Connecticut author notesSection 2, Theorem 2.1, printed page 1; uniqueness proof on printed page 3 · Checked 2026-10-14

Translation sources

1. 张神星 — 代数数论讲义 v2.5 (2024-01-09)，合肥工业大学作者讲义引理2.15 亨泽尔引理，printed page 37 · Checked 2026-10-14

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