Hensel's lemma

From Wikipedia, the free encyclopedia

In mathematics, Hensel's lemma, named after Kurt Hensel, is 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.

Contents

[edit] First form

A version of the lemma for p-adic fields is as follows. Let f(x) be a polynomial with integer coefficients, k an integer not less than 2 and p a prime number. Suppose that r is a solution of the congruence

f(r) \equiv 0 \pmod{p^{k-1}}.\,

If f'(r) \not\equiv 0 \pmod{p}, then there is a unique integer t, 0 ≤ t ≤ p-1, such that

f(r + tp^{k-1}) \equiv 0 \pmod{p^k}\,

with t defined by

tf'(r) \equiv -(f(r)/p^{k-1}) \pmod{p}.\,

If, on the other hand, f'(r) \equiv 0 \pmod{p}, and in addition, f(r) \equiv 0 \pmod{p^k}, then

f(r + tp^{k-1}) \equiv 0 \pmod{p^k}\,

for all integers t.

Also, if f'(r) \equiv 0 \pmod{p}\, and f(r) \not\equiv 0 \pmod{p^k}, then f(x) \equiv 0 \pmod{p^k}\, has no solution for any x \equiv r \pmod{p^{k-1}}.\,

[edit] Example

Suppose that p is an odd prime number and a is a quadratic residue modulo p relative prime to p. Then a has a square root in the ring of p-adic integers Zp. Indeed, let f(x)=x2-a, then its derivative is 2x, which is not zero modulo p for x not divisible by p (here we use that p is odd). By the assumption, the congruence

f(r)\equiv 0 \pmod{p}

has a solution r1 not divisible by p. Starting from r1 and repeatedly applying Hensel's lemma, we construct a sequence of integers { ri } such that

r_{i+1} \equiv r_i \pmod{p^i}, \quad r_i^2 \equiv a \pmod{p^i}.

This sequence has a limit, a p-adic integer r such that r2=a. In fact, r is a unique square root of a in Zp congruent to r1 modulo p. Conversely, if a is a complete square in Zp then it is a quadratic residue mod p. Note that the Quadratic reciprocity law allows one to easily test whether a is a quadratic residue mod p, thus we get a practical way to determine which p-adic numbers (for p odd) have an integral square root, and it can be easily extended to cover the case p=2.

To make the example more explicit, let us consider finding the "square root of 2" (the solution to x2 − 2 = 0) in the 7-adic integers. Modulo 7, we have a solution, namely 3 (we could also take 4), so r1 = 3. Hensel's lemma then allows us to find r2 as follows:

f(r1) = 32 − 2 = 7
f(r1) / p1 = 7 / 7 = 1
f'(r1) = 2r1 = 6
tf'(r) \equiv -(f(r)/p^{k-1}) \pmod{p}, that is, t\cdot 6 \equiv -1 \pmod{7}
\Rightarrow t = 1
r_2 = r_1 + tp^1 = 3+1 \cdot 7 = 10 =13_7

And sure enough, 10^2\equiv 2 \pmod{49}. We can then continue, and find r3 = 108 = 2137. Each time we carry out the calculation (that is, for each successive value of k or i), one more digit is added on the left to the base 7 numeral. In the 7-adic system, this sequence converges, and the limit is thus a square root of 2 in this field.

[edit] Generalizations

Suppose A is a commutative ring, complete with respect to an ideal \mathfrak m_A, and let f(x) \in A[x] be a polynomial with coefficients in A. Then if a ∈ A is an "approximate root" of f in the sense that it satisfies

f(a) \equiv 0 \pmod{f'(a)^2 \,\mathfrak m}

then there is an exact root b ∈ A of f "close to" a; that is,

f(b) = 0

and

b \equiv a \pmod{f'(a) \,\mathfrak m}.

Further, if f ′(a) is not a zero-divisor then b is unique.

This result has been generalized to several variables by Nicolas Bourbaki as follows:

Theorem: Let A be a commutative ring, complete with respect to an ideal m⊂ A (which is equivalent to the fact that there is an absolute value on A so that for every x in m we have |x| is strictly less than 1 and the resulting metric space is complete), and a = (a1, …, an) ∈ An an approximate solution to a system of polynomials fi(x) ∈ A[x1, …, xn] in the sense that

fi(a) ≡ 0 mod m

for 1 ≤ i ≤ n. Suppose that either det J(a) is a unit in A or that each fi(a) ∈ (det J(a))²m, where J(a) is the Jacobian matrix of a with respect to the fi. Then there is an exact solution b = (b1, …, bn) in the sense that

fi(b) = 0

and furthermore this solution is "close to" a in the sense that

bi ≡ ai mod m

for 1 ≤ i ≤ n.

[edit] 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.

[edit] See also

[edit] References