# Hensel's lemma

**Hensel's lemma** is a theorem of number theory stating that a simple root of a polynomial modulo a prime p lifts, uniquely, to a genuine root in the ring of p-adic integers (number system completing integers relative to a prime), and more generally that a factorization of a polynomial into coprime factors modulo p lifts to a factorization over the p-adics. [Kurt Hensel](https://www.edgepedia.org/kurt-hensel) obtained the statement while creating the theory of p-adic numbers, and it has since become one of the standard tools for solving polynomial equations over local fields.<sup>[2](https://encyclopediaofmath.org/index.php?title=Hensel_lemma)</sup><sup> • </sup><sup>[6](https://www.exhibit.xavier.edu/oresme_2018Oct/2/)</sup>

| Key fact | Detail |
|---|---|
| Root-lifting form | If \( f \in \mathbb{Z}_p[X] \) and \( \lvert f(a)\rvert_p < \lvert f'(a)\rvert_p^{2} \), there is a unique \( \alpha \in \mathbb{Z}_p \) with \( f(\alpha)=0 \) and \( \lvert \alpha-a\rvert_p < \lvert f'(a)\rvert_p \)<sup>[1](https://web.ma.utexas.edu/users/sl55444/M328KNotes.pdf)</sup> |
| Factor-lifting form | For a complete local ring \( A \) with maximal ideal \( \mathfrak m \), a monic \( f \) whose reduction factors into coprime monic \( \bar g, \bar h \) factors over \( A \) with factors reducing to \( \bar g, \bar h \)<sup>[3](https://dkatz.ku.edu/Math%20996/hensel.pdf)</sup> |
| Method of proof | Newton iteration \( x_{n+1} = x_n - f(x_n)/f'(x_n) \); the error estimates are proved by induction using Taylor expansion<sup>[1](https://web.ma.utexas.edu/users/sl55444/M328KNotes.pdf)</sup> |
| Standard example | \( X^2-2=0 \) is solvable in \( \mathbb{Z}_7 \) because it has two simple roots in \( \mathbb{F}_7 \)<sup>[2](https://encyclopediaofmath.org/index.php?title=Hensel_lemma)</sup> |
| Generality | Holds for any complete discrete valuation ring; a local ring for which the lemma holds is called Henselian<sup>[10](https://link.springer.com/chapter/10.1007/978-3-030-56694-4_12)</sup> |
| Origin | Kurt Hensel introduced the p-adic numbers in 1897 and developed the theory in *Theorie der algebraischen Zahlen* (1908)<sup>[6](https://www.exhibit.xavier.edu/oresme_2018Oct/2/)</sup><sup> • </sup><sup>[7](https://mathshistory.st-andrews.ac.uk//Biographies/Hensel/)</sup> |
| Computational role | Hensel lifting is a core component of polynomial factorization algorithms such as Zassenhaus's<sup>[13](https://www.cambridge.org/core/books/abs/modern-computer-algebra/hensel-lifting-and-factoring-polynomials/93C3F743F3022F2A085AC52020365460)</sup> |

## Statement and meaning

The most common form concerns the ring \( \mathbb{Z}_p \) of p-adic integers. Let \( f(X) \) be a polynomial over \( \mathbb{Z}_p \) and suppose an element \( a \in \mathbb{Z}_p \) satisfies

\[ \lvert f(a) \rvert_p < \lvert f'(a) \rvert_p^{2}. \]

Then there exists a unique \( \alpha \in \mathbb{Z}_p \) such that \( f(\alpha)=0 \) and \( \lvert \alpha-a \rvert_p < \lvert f'(a) \rvert_p \).<sup>[1](https://web.ma.utexas.edu/users/sl55444/M328KNotes.pdf)</sup> In concrete congruence terms, a solution of \( f(x) \equiv 0 \pmod p \) at which the derivative is not divisible by p extends to a solution modulo \( p^n \) for every n, and these compatible solutions assemble into an actual p-adic root.<sup>[11](https://math.uchicago.edu/~may/REU2016/REUPapers/Kirsche.pdf)</sup> The derivative condition says exactly that the root modulo p is simple; a multiple root modulo p cannot in general be lifted uniquely.<sup>[1](https://web.ma.utexas.edu/users/sl55444/M328KNotes.pdf)</sup>

A second, factorization form generalizes the root case. Let \( (A, \mathfrak m, k) \) be a local ring complete in its \( \mathfrak m \)-adic topology, and let \( f(X) \in A[X] \) be monic of degree d. If \( \bar f = \bar g \cdot \bar h \) in \( k[X] \) with \( \bar g \) and \( \bar h \) monic and relatively prime, of degrees r and d−r, then \( f \) itself factors as \( \hat g \cdot \hat h \) with monic factors of those degrees reducing to \( \bar g \) and \( \bar h \).<sup>[3](https://dkatz.ku.edu/Math%20996/hensel.pdf)</sup> Taking \( \bar g \) linear recovers the root-lifting statement: a simple root \( \alpha \) of \( \bar f \) comes from a unique \( a \in A \) with \( f(a)=0 \).<sup>[2](https://encyclopediaofmath.org/index.php?title=Hensel_lemma)</sup><sup> • </sup><sup>[3](https://dkatz.ku.edu/Math%20996/hensel.pdf)</sup>

## Proof by Newton iteration

The proof runs [Newton's method](https://www.edgepedia.org/newtons-method) from the approximate root \( a \), defining \( x_0 = a \) and

\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}. \]

Writing \( \delta = \lvert f(a) \rvert_p / \lvert f'(a) \rvert_p^{2} < 1 \), one proves by induction, using Taylor expansion, the estimates \( \lvert x_{n+1}-x_n \rvert_p \le \lvert f'(a) \rvert_p \, \delta^{2^n} \) and \( \lvert f(x_n) \rvert_p \le \lvert f'(a) \rvert_p^{2} \, \delta^{2^n} \). The sequence is therefore Cauchy, and its limit \( \alpha \) satisfies \( f(\alpha)=0 \); uniqueness follows from the ultrametric triangle inequality together with a Taylor expansion argument.<sup>[1](https://web.ma.utexas.edu/users/sl55444/M328KNotes.pdf)</sup> Because each step roughly squares the accuracy, the correct digits double at every iteration, the same quadratic convergence Newton's method shows over the reals.<sup>[1](https://web.ma.utexas.edu/users/sl55444/M328KNotes.pdf)</sup> The same iteration proves a version over any field complete for a non-archimedean absolute value: if \( \lvert f(\alpha_0) \rvert < \lvert f'(\alpha_0) \rvert^{2} \), the Newton sequence converges to a root \( \alpha \) with \( \lvert \alpha - \alpha_0 \rvert \le \lvert f(\alpha_0)/f'(\alpha_0)^2 \rvert \).<sup>[4](https://www.math.toronto.edu/~ila/ANTBook.pdf)</sup>

## Example: a square root of 2 in the 7-adics

The congruence \( x^2 \equiv 2 \pmod 7 \) has the solutions 3 and 4, both simple because \( 2x \not\equiv 0 \pmod 7 \) at them. Hensel's lemma therefore yields two square roots of 2 in \( \mathbb{Z}_7 \).<sup>[2](https://encyclopediaofmath.org/index.php?title=Hensel_lemma)</sup> The 1909 Nature review of Hensel's book describes the lifting concretely: writing the successive digits in the septenary scale gives the sequence 3, 31, 312, 3126, ..., where each truncation \( a_1 a_2 \cdots a_n \) solves \( x^2 \equiv 2 \pmod{7^n} \), and the infinite digit string is the septenary representation of \( \sqrt{2} \) in Hensel's extended sense of number.<sup>[9](https://www.nature.com/articles/082095a0)</sup>

## Applications

**Roots of unity.** Since \( x^p - x \) splits into distinct linear factors over \( \mathbb{F}_p \) and has derivative \(-1\), Hensel's lemma shows that \( \mathbb{Z}_p \) contains \( p-1 \) distinct \( (p-1) \)-st roots of unity, one in each nonzero residue class modulo p.<sup>[4](https://www.math.toronto.edu/~ila/ANTBook.pdf)</sup>

**Local solutions of equations.** The lemma is described as one of the most useful tools for finding zeros of polynomials over the p-adics: a nontrivial solution of a polynomial equation modulo p with nonvanishing derivative lifts to a full p-adic solution, a step used for instance in verifying local solvability of quadratic forms.<sup>[11](https://math.uchicago.edu/~may/REU2016/REUPapers/Kirsche.pdf)</sup>

**Factorization and computer algebra.** Hensel lifting, the algorithmic form of the lemma, raises a factorization modulo p to one modulo higher powers of p. It is a central component of the Zassenhaus factorization algorithm for polynomials over the rationals and of its modern successors, where factors found over a finite field are lifted and then recombined over the integers.<sup>[13](https://www.cambridge.org/core/books/abs/modern-computer-algebra/hensel-lifting-and-factoring-polynomials/93C3F743F3022F2A085AC52020365460)</sup> Recent work extends the lemma to generalized Newton polygons to speed local factorization over complete discrete valuation rings.<sup>[12](https://arxiv.org/abs/2607.02153)</sup>

## Generalizations and formalization

The lemma holds, with essentially the same proof, for any complete discrete valuation ring, without conditions on the residue field; abstracting this property defines Henselian rings, local rings for which Hensel's lemma is valid.<sup>[10](https://link.springer.com/chapter/10.1007/978-3-030-56694-4_12)</sup><sup> • </sup><sup>[2](https://encyclopediaofmath.org/index.php?title=Hensel_lemma)</sup> Weaker analogs hold over other rings such as the ordinary integers, though their hypotheses are harder to satisfy than those over \( \mathbb{Z}_p \).<sup>[5](https://arxiv.org/abs/1909.11342)</sup> Versions also exist for analytic functions on the p-adic integers.<sup>[14](https://www.sciencedirect.com/science/article/pii/S0022314X85710566)</sup>

The theorem has been machine-verified. Robert Y. Lewis, a researcher in formalized mathematics, constructed \( \mathbb{Q}_p \) and \( \mathbb{Z}_p \) in the Lean proof assistant and proved a strong form of Hensel's lemma there in 2019, following a writeup by Keith Conrad, a mathematician at the [University of Connecticut](https://www.edgepedia.org/university-of-connecticut); the Lean documentation describes the lemma, citing Fernando Gouvêa's 1997 textbook, as the "most important algebraic property of the p-adic numbers."<sup>[5](https://arxiv.org/abs/1909.11342)</sup><sup> • </sup><sup>[8](https://leanprover-community.github.io/theories/padics.html)</sup> An Isabelle/HOL formalization of the p-adic integers in the Archive of Formal Proofs includes Hensel's lemma based on the same proof.<sup>[15](https://isa-afp.org/entries/Padic_Ints.html)</sup>

## History

Kurt Hensel (1861–1941), a student of Leopold Kronecker, introduced the p-adic numbers in 1897, motivated by the analogy, pointed out by Kronecker and by [Richard Dedekind](https://www.edgepedia.org/richard-dedekind) and Heinrich Weber, between prime numbers and the points of a [Riemann surface](https://www.edgepedia.org/riemann-surface); the Weierstrass method of power-series development for algebraic functions led him to expand algebraic numbers in p-adic power series.<sup>[7](https://mathshistory.st-andrews.ac.uk//Biographies/Hensel/)</sup> He developed the theory systematically in *Theorie der algebraischen Zahlen* (Teubner, 1908) and *Zahlentheorie* (1913), using the lemma that now bears his name to lift polynomial factorizations modulo p or \( p^k \) to p-adic factorizations.<sup>[6](https://www.exhibit.xavier.edu/oresme_2018Oct/2/)</sup><sup> • </sup><sup>[7](https://mathshistory.st-andrews.ac.uk//Biographies/Hensel/)</sup> The full power of the p-adic viewpoint became visible in 1921, when Hensel's student [Helmut Hasse](https://www.edgepedia.org/helmut-hasse) formulated the local-global principle, proving for quadratic forms that a rational solution exists exactly when solutions exist in the reals and in every \( \mathbb{Q}_p \).<sup>[7](https://mathshistory.st-andrews.ac.uk//Biographies/Hensel/)</sup>

## References

1. [Introduction to Number Theory, M328K notes, University of Texas (Hensel's lemma, nice version)](https://web.ma.utexas.edu/users/sl55444/M328KNotes.pdf)
2. [Hensel lemma, Encyclopedia of Mathematics](https://encyclopediaofmath.org/index.php?title=Hensel_lemma)
3. [D. Katz, Hensel's Lemma and Some Applications](https://dkatz.ku.edu/Math%20996/hensel.pdf)
4. [I. Varma, Algebraic Number Theory, University of Toronto](https://www.math.toronto.edu/~ila/ANTBook.pdf)
5. [R. Y. Lewis (2019), A Formal Proof of Hensel's Lemma over the p-adic Integers, arXiv](https://arxiv.org/abs/1909.11342)
6. [Hensel, Kurt, "Theorie Der Algebraischen Zahlen" (1908), Xavier University exhibit](https://www.exhibit.xavier.edu/oresme_2018Oct/2/)
7. [Kurt Hensel (1861–1941), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Biographies/Hensel/)
8. [Maths in Lean: p-adic numbers, Lean prover community](https://leanprover-community.github.io/theories/padics.html)
9. [G. M., Review of Hensel, Theorie der algebraischen Zahlen, Nature 82, 95–96 (1909)](https://www.nature.com/articles/082095a0)
10. [Ternary quadratic forms over local fields, Springer](https://link.springer.com/chapter/10.1007/978-3-030-56694-4_12)
11. [Z. Kirsche, Local-Global Methods in Algebraic Number Theory, University of Chicago REU](https://math.uchicago.edu/~may/REU2016/REUPapers/Kirsche.pdf)
12. [A. Poteaux, M. Weimann, Local polynomial factorisation: improving the Montes algorithm, arXiv](https://arxiv.org/abs/2607.02153)
13. [Hensel lifting and factoring polynomials, Modern Computer Algebra, Cambridge University Press](https://www.cambridge.org/core/books/abs/modern-computer-algebra/hensel-lifting-and-factoring-polynomials/93C3F743F3022F2A085AC52020365460)
14. [Hensel's Lemma and the Divisibility by Primes of Stirling-like Numbers, Journal of Number Theory 52 (1995)](https://www.sciencedirect.com/science/article/pii/S0022314X85710566)
15. [Hensel's Lemma for the p-adic Integers, Archive of Formal Proofs](https://isa-afp.org/entries/Padic_Ints.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Number systems › p-adic numbers › p-adic methods in number theory*

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