# Hensel's lemma

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In mathematics, **Hensel's lemma**, also known as **Hensel's lifting lemma**, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number *p*, then this root can be *lifted* to a unique root modulo any higher power of *p*. More generally, if a polynomial factors modulo *p* into two coprime polynomials, this factorization can be lifted to a factorization modulo any higher power of *p* (the case of roots corresponds to the case of degree 1 for one of the factors).

By passing to the "limit" (in fact this is an inverse limit) when the power of p tends to infinity, it follows that a root or a factorization modulo p can be lifted to a root or a factorization over the p-adic integers.

These results have been widely generalized, under the same name, to the case of polynomials over an arbitrary commutative ring, where p is replaced by an ideal, and "coprime polynomials" means "polynomials that generate an ideal containing 1".

Hensel's lemma is fundamental in p-adic analysis, a branch of analytic number theory.

The proof of Hensel's lemma is constructive, and leads to an efficient algorithm for **Hensel lifting**, which is fundamental for factoring polynomials, and gives the most efficient known algorithm for exact linear algebra over the rational numbers.