Hensel's lemma
Hensel's lemma
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 polynomial equation has a simple root modulo a prime number p, then this root corresponds to a unique root of the same equation modulo any higher power of p, which can be found by iteratively "lifting" the solution modulo successive powers of p. More generally it is used as a generic name for analogues for complete commutative rings (including p-adic fields in particular) of the Newton method for solving equations. Since p-adic analysis is in some ways simpler than real analysis, there are relatively neat criteria guaranteeing a root of a polynomial.
Statement
Many equivalent statements of Hensel's lemma exist. Arguably the most common statement is the following.
General statement
Alternative statement
then there exists an integer s such that
Furthermore, this s is unique modulo p**k+m, and can be computed explicitly as the integer such that
Derivation
We use the Taylor expansion of f around r to write:
- for some integer t. Let
Hensel lifting
- to a new root s modulo p
- (by taking m=1; taking larger m follows by induction). In fact, a root modulo p
- of
- as mod p
- are all simple.
What happens to this process if r is not a simple root mod p? Suppose
If then there is no lifting of r to a root of f(x) modulo p**k+1.
If then every lifting of r to modulus p**k+1 is a root of f(x) modulo p**k+1.
Example. To see both cases we examine two different polynomials with p = 2:
Hensel's lemma for p-adic numbers
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 r**k+1 (roots mod p**k+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):
Then set
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 r**k+1 in terms of rk is precisely Newton's method for finding roots to equations in the real numbers.
Examples
Based on which the expression
turns into:
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.
- 1 mod 2 --> 1, 3 mod 41 mod 4 --> 1, 5 mod 8 and 3 mod 4 ---> 3, 7 mod 81 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 169 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 + 23and 25 = 1 + 23
- 2
- 2
The 2-adic square roots of 17 have expansions
Generalizations
If f has an approximate root then it has an exact root b ∈ A "close to" a; that is,
This result can be generalized to several variables as follows:
- Theorem. Suppose A be a commutative ring that is complete with respect to idealLetbe a system of n polynomials in n variables over A. Viewas a mapping from *An
- to itself, and let
- 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
Related concepts
Completeness of a ring is not a necessary condition for the ring to have the Henselian property: Goro Azumaya in 1950 defined a commutative local ring satisfying the Henselian property for the maximal ideal m to be a Henselian ring.
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  while still retaining the Henselian property and remaining in the same category.
See also
Hasse–Minkowski theorem
Newton polygon