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 numberp, 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 original lemma concerns the relation between polynomial factorization over the integers and over the integers modulo a prime numberp 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 have the form 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 moduloI refers to the replacement of every element of R by its image under the canonical map For example, if is a polynomial with coefficients in R, its reduction modulo I, denoted is the polynomial in obtained by replacing the coefficients of f by their image in Two polynomials f and g in are congruent moduloI, denoted if they have the same coefficients modulo I, that is if If a factorization of h modulo I consists in two (or more) polynomials f, g in such that
The lifting process is the inverse of reduction. That is, given objects depending on elements of the lifting process replaces these elements by elements of (or of for some k > 1) that maps to them in a way that keeps the properties of the objects.
For example, given a polynomial and a factorization modulo I expressed as lifting this factorization modulo consists of finding polynomials such that and Hensel's lemma asserts that such a lifting is always possible under mild conditions; see next section.
Originally, Hensel's lemma was stated (and proved) for lifting a factorization modulo a prime numberp 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 be a maximal ideal of a commutative ring R, and
An important special case is when In this case the coprimality hypothesis means that r is a simple root of This gives the following special case of Hensel's lemma, which is often called also Hensel's lemma.
With above hypotheses and notations, if r is a simple root of then r can be lifted in a unique way to a simple root of for every positive integer n. Explicitly, for every positive integer n, there is a unique such that and is a simple root of
The definition of the completion as an inverse limit, and the above statement of Hensel's lemma imply that every factorization into pairwise coprime polynomials modulo of a polynomial can be uniquely lifted to a factorization of the image of h in Similarly, every simple root of h modulo can be lifted to a simple root of the image of h in
Bézout's identity allows defining coprime polynomials and proving Hensel's lemma, even if the ideal is not maximal. Therefore, in the following proofs, one starts from a commutative ring R, an idealI, a polynomial that has a leading coefficient that is invertible modulo I (that is its image in is a unit in ), and factorization of h modulo I or modulo a power of I, such that the factors satisfy a Bézout's identity modulo I. In these proofs, means
Let R, I, h and as a in the preceding section. Let
be a factorization into coprime polynomials (in the above sense), such The application of linear lifting for shows the existence of and such that and
The polynomials and are uniquely defined modulo This means that, if another pair satisfies the same conditions, then one has
Proof: Since a congruence modulo implies the same concruence modulo one can proceed by induction and suppose that the uniqueness has been proved for n − 1, the case n = 0 being trivial. That is, one can suppose that
By hypothesis, has
By induction hypothesis, the second term of the latter sum belongs to and the same is thus true for the first term. As is invertible modulo I, there exist and such that Thus
using the induction hypothesis again.
The coprimality modulo I implies the existence of such that Using the induction hypothesis once more, one gets
Thus one has a polynomial of degree less than that is congruent modulo to the product of the monic polynomial g and another polynomial w. This is possible only if and implies Similarly, is also in and this proves the uniqueness.
Linear lifting allows lifting a factorization modulo to a factorization modulo Quadratic lifting allows lifting directly to a factorization modulo at the cost of lifting also the Bézout's identity and of computing modulo instead of modulo I (if one uses the above description of linear lifting).
For lifting up to modulo for large N one can use either method. If, say, a factorization modulo requires N − 1 steps of linear lifting or only k − 1 steps of quadratic lifting. However, in the latter case the size of the coefficients that have to be manipulated increase during the computation. This implies that the best lifting method depends on the context (value of N, nature of R, multiplication algorithm that is used, hardware specificities, etc.).
Quadratic lifting is based on the following property.
Suppose that for some positive integer k there is a factorization
such that f and g are monic polynomials that are coprime modulo I, in the sense that there exist such that Then, there are polynomials such that and
Moreover, and satisfy a Bézout's identity of the form
(This is required for allowing iterations of quadratic lifting.)
Proof: The first assertion is exactly that of linear lifting applied with k = 1 to the ideal instead of I.
Using the above hypotheses, if we consider an irreducible polynomial
such that , then
In particular, for , we find in
but , hence the polynomial cannot be irreducible. Whereas in we have both values agreeing, meaning the polynomial could be irreducible. In order to determine irreducibility, the Newton polygon must be employed.pg 144
Although the -th roots of unity are not contained in , there are solutions of . Note
is never zero, so if there exists a solution, it necessarily lifts to . Because the Frobenius gives all of the non-zero elements are solutions. In fact, these are the only roots of unity contained in .
Using the lemma, one can "lift" a root r of the polynomial f modulo pk to a new root s modulo pk+1 such that r ≡ s mod pk (by taking m=1; taking larger m follows by induction). In fact, a root modulo pk+1 is also a root modulo pk, so the roots modulo pk+1 are precisely the liftings of roots modulo pk. The new root s is congruent to r modulo p, so the new root also satisfies So the lifting can be repeated, and starting from a solution rk of we can derive a sequence of solutions rk+1, rk+2, ... of the same congruence for successively higher powers of p, provided for the initial root rk. This also shows that f has the same number of roots mod pk as mod pk+1, mod pk+2, or any other higher power of p, provided the roots of f mod pk are all simple.
What happens to this process if r is not a simple root mod p? Suppose
Then implies That is, for all integers t. Therefore, we have two cases:
If then there is no lifting of r to a root of f(x) modulo pk+1.
If then every lifting of r to modulus pk+1 is a root of f(x) modulo pk+1.
Example. To see both cases we examine two different polynomials with p = 2:
and r = 1. Then and We have which means that no lifting of 1 to modulus 4 is a root of f(x) modulo 4.
and r = 1. Then and However, since we can lift our solution to modulus 4 and both lifts (i.e. 1, 3) are solutions. The derivative is still 0 modulo 2, so a priori we don't know whether we can lift them to modulo 8, but in fact we can, since g(1) is 0 mod 8 and g(3) is 0 mod 8, giving solutions at 1, 3, 5, and 7 mod 8. Since of these only g(1) and g(7) are 0 mod 16 we can lift only 1 and 7 to modulo 16, giving 1, 7, 9, and 15 mod 16. Of these, only 7 and 9 give g(x) = 0 mod 32, so these can be raised giving 7, 9, 23, and 25 mod 32. It turns out that for every integer k ≥ 3, there are four liftings of 1 mod 2 to a root of g(x) mod 2k.
In the p-adic numbers, where we can make sense of rational numbers modulo powers of p as long as the denominator is not a multiple of p, the recursion from rk (roots mod pk) to rk+1 (roots mod pk+1) can be expressed in a much more intuitive way. Instead of choosing t to be an(y) integer which solves the congruence
let t be the rational number (the pk here is not really a denominator since f(rk) is divisible by pk):
This fraction may not be an integer, but it is a p-adic integer, and the sequence of numbers rk converges in the p-adic integers to a root of f(x) = 0. Moreover, the displayed recursive formula for the (new) number rk+1 in terms of rk is precisely Newton's method for finding roots to equations in the real numbers.
By working directly in the p-adics and using the p-adic absolute value, there is a version of Hensel's lemma which can be applied even if we start with a solution of f(a) ≡ 0 mod p such that We just need to make sure the number is not exactly 0. This more general version is as follows: if there is an integer a which satisfies:
then there is a unique p-adic integer b such f(b) = 0 and The construction of b amounts to showing that the recursion from Newton's method with initial value a converges in the p-adics and we let b be the limit. The uniqueness of b as a root fitting the condition needs additional work.
The statement of Hensel's lemma given above (taking ) is a special case of this more general version, since the conditions that f(a) ≡ 0 mod p and say that and
Suppose that p is an odd prime and a is a non-zero quadratic residue modulo p. Then Hensel's lemma implies that a has a square root in the ring of p-adic integers Indeed, let If r is a square root of a modulo p then:
where the second condition is dependent on the fact that p is odd. The basic version of Hensel's lemma tells us that starting from r1 = r we can recursively construct a sequence of integers such that:
This sequence converges to some p-adic integer b which satisfies b2 = a. In fact, b is the unique square root of a in congruent to r1 modulo p. Conversely, if a is a perfect square in and it is not divisible by p then it is a nonzero quadratic residue mod p. Note that the quadratic reciprocity law allows one to easily test whether a is a nonzero quadratic residue mod p, thus we get a practical way to determine which p-adic numbers (for p odd) have a p-adic square root, and it can be extended to cover the case p = 2 using the more general version of Hensel's lemma (an example with 2-adic square roots of 17 is given later).
To make the discussion above more explicit, let us find a "square root of 2" (the solution to ) in the 7-adic integers. Modulo 7 one solution is 3 (we could also take 4), so we set . Hensel's lemma then allows us to find as follows:
Based on which the expression
which implies Now:
And sure enough, (If we had used the Newton method recursion directly in the 7-adics, then and )
We can continue and find . Each time we carry out the calculation (that is, for each successive value of k), one more base 7 digit is added for the next higher power of 7. In the 7-adic integers this sequence converges, and the limit is a square root of 2 in which has initial 7-adic expansion
If we started with the initial choice then Hensel's lemma would produce a square root of 2 in which is congruent to 4 (mod 7) instead of 3 (mod 7) and in fact this second square root would be the negative of the first square root (which is consistent with 4 = −3 mod 7).
As an example where the original version of Hensel's lemma is not valid but the more general one is, let and Then and so
which implies there is a unique 2-adic integer b satisfying
i.e., b ≡ 1 mod 4. There are two square roots of 17 in the 2-adic integers, differing by a sign, and although they are congruent mod 2 they are not congruent mod 4. This is consistent with the general version of Hensel's lemma only giving us a unique 2-adic square root of 17 that is congruent to 1 mod 4 rather than mod 2. If we had started with the initial approximate root a = 3 then we could apply the more general Hensel's lemma again to find a unique 2-adic square root of 17 which is congruent to 3 mod 4. This is the other 2-adic square root of 17.
In terms of lifting the roots of from modulus 2k to 2k+1, the lifts starting with the root 1 mod 2 are as follows:
1 mod 2 --> 1, 3 mod 4
1 mod 4 --> 1, 5 mod 8 and 3 mod 4 ---> 3, 7 mod 8
1 mod 8 --> 1, 9 mod 16 and 7 mod 8 ---> 7, 15 mod 16, while 3 mod 8 and 5 mod 8 don't lift to roots mod 16
9 mod 16 --> 9, 25 mod 32 and 7 mod 16 --> 7, 23 mod 16, while 1 mod 16 and 15 mod 16 don't lift to roots mod 32.
For every k at least 3, there are four roots of x2 − 17 mod 2k, but if we look at their 2-adic expansions we can see that in pairs they are converging to just two 2-adic limits. For instance, the four roots mod 32 break up into two pairs of roots which each look the same mod 16:
9 = 1 + 23 and 25 = 1 + 23 + 24.
7 = 1 + 2 + 22 and 23 = 1 + 2 + 22 + 24.
The 2-adic square roots of 17 have expansions
Another example where we can use the more general version of Hensel's lemma but not the basic version is a proof that any 3-adic integer c ≡ 1 mod 9 is a cube in Let and take initial approximation a = 1. The basic Hensel's lemma cannot be used to find roots of f(x) since for every r. To apply the general version of Hensel's lemma we want which means That is, if c ≡ 1 mod 27 then the general Hensel's lemma tells us f(x) has a 3-adic root, so c is a 3-adic cube. However, we wanted to have this result under the weaker condition that c ≡ 1 mod 9. If c ≡ 1 mod 9 then c ≡ 1, 10, or 19 mod 27. We can apply the general Hensel's lemma three times depending on the value of c mod 27: if c ≡ 1 mod 27 then use a = 1, if c ≡ 10 mod 27 then use a = 4 (since 4 is a root of f(x) mod 27), and if c ≡ 19 mod 27 then use a = 7. (It is not true that every c ≡ 1 mod 3 is a 3-adic cube, e.g., 4 is not a 3-adic cube since it is not a cube mod 9.)
In a similar way, after some preliminary work, Hensel's lemma can be used to show that for any odd prime number p, any p-adic integer c congruent to 1 modulo p2 is a p-th power in (This is false for p = 2.)
If f has an approximate root then it has an exact root b ∈ A "close to" a; that is,
Furthermore, if is not a zero-divisor then b is unique.
This result can be generalized to several variables as follows:
Theorem. Let A be a commutative ring that is complete with respect to ideal Let be a system of n polynomials in n variables over A. View as a mapping from An to itself, and let denote its Jacobian matrix. Suppose a = (a1, ..., an) ∈ An is an approximate solution to f = 0 in the sense that
Then there is some b = (b1, ..., bn) ∈ An satisfying f(b) = 0, i.e.,
Furthermore this solution is "close" to a in the sense that
As a special case, if for all i and is a unit in A then there is a solution to f(b) = 0 with for all i.
When n = 1, a = a is an element of A and The hypotheses of this multivariable Hensel's lemma reduce to the ones which were stated in the one-variable Hensel's lemma.
Masayoshi Nagata proved in the 1950s that for any commutative local ring A with maximal ideal m there always exists a smallest ring Ah containing A such that Ah is Henselian with respect to mAh. This Ah is called the Henselization of A. If A is noetherian, Ah will also be noetherian, and Ah is manifestly algebraic as it is constructed as a limit of étale neighbourhoods. This means that Ah is usually much smaller than the completion Â while still retaining the Henselian property and remaining in the same category[clarification needed].