亨泽尔引理

定理内容

设 f(x) 为整系数多项式， k 为不少于2的整数， p 为质数。若整数 r 是下面同余式的根： （公式略）

对于 （公式略） （I）

，则有：

若 （公式略），则存在唯一的整数 0≤ t≤ p-1 使得（I）成立。 （公式略）

若 （公式略） 且 （公式略），则（I）对任意整数t成立。

若 （公式略） 但 （公式略），则（I）无整数解。

证明

亨泽尔引理可用泰勒公式证明。 （公式略）

因此可见，由第三项开始，都必能被 p^(k) 整除。因此： （公式略）

推广

若 K 为完备局域。设 O_(K) 为 K 的整数环，设 f(x) 为系数在 O_(K) 的多项式，若存在 α ₀∈ O_(K) 使得 |f(α ₀)|<|f'(α ₀)|²

则 f(x) 有根 α ∈ K。

且： （公式略） 趋近 α

| （公式略）

这个引理其中一个重要应用就是在域为p进数的情形。

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. (The lifting process is the inverse of the reduction modulo p and is precisely defined below.) 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.

Modular reduction and lifting

Hensel's original lemma concerns the relation between polynomial factorization over the integers and over the integers modulo a prime number p and its powers. It can be straightforwardly extended to the case where the integers are replaced by any commutative ring, and p is replaced by any maximal ideal (indeed, the maximal ideals of Z have the form pZ, where p is a prime number).

Making this precise requires a generalization of the usual modular arithmetic, and so it is useful to define accurately the terminology that is commonly used in this context.

Let R be a commutative ring, and I an ideal of R. Reduction modulo I refers to the replacement of every element of R by its image under the canonical map R→ R/I.

Statement

Originally, Hensel's lemma was stated (and proved) for lifting a factorization modulo a prime number p of a polynomial over the integers to a factorization modulo any power of p and to a factorization over the p-adic integers. This can be generalized easily, with the same proof to the case where the integers are replaced by any commutative ring, the prime number is replaced by a maximal ideal, and the p-adic integers are replaced by the completion with respect to the maximal ideal. It is this generalization, which is also widely used, that is presented here.

Let （公式略） be a maximal ideal of a commutative ring R, and h=α ₀Xⁿ+⋯ +α ₙ₋₁X+α ₙ

be a polynomial in R[X] with a leading coefficient α ₀ not in （公式略）

Proof

Hensel's lemma is generally proved incrementally by lifting a factorization over （公式略） to either a factorization over （公式略） (Linear lifting), or a factorization over （公式略） (Quadratic lifting).

The main ingredient of the proof is that coprime polynomials over a field satisfy Bézout's identity. That is, if f and g are coprime univariate polynomials over a field (here （公式略）), there are polynomials a and b such that deg a<deg g,

deg deg b<deg f, and af+bg=1.

Bézout's identity allows defining coprime polynomials and proving Hensel's lemma, even if the ideal （公式略） is not maximal.

Explicit example

Let f(X)=X⁶-2∈ Q [X].

Modulo 2, Hensel's lemma cannot be applied since the reduction of f(X) modulo 2 is simplypg 15-16 f(X)=X⁶-2=X⁶

with 6 factors X not being relatively prime to each other. By Eisenstein's criterion, however, one can conclude that the polynomial f(X) is irreducible in Q ₂[X].

Over k=F ₇, on the other hand, one has f(X)=X⁶-2=X⁶-16=(X³-4) (X³+4)

where 4 is the square root of 2 in F ₇. As 4 is not a cube in F ₇, these two factors are irreducible over F ₇. Hence the complete factorization of X⁶-2 Z ₇[X] and Q ₇[X] f(X)=X⁶-2=(X³-α) (X³+α),

where α =… 450 454₇ is a square root of 2 in Z ₇ that can be obtained by lifting the above factorization.

Finally, in F ₇₂₇[X] the polynomial splits into f(X)=X⁶-2=(X-3) (X-116) (X-119) (X-608) (X-611) (X-724)