Hensel's lemma

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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.

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

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

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

with t given by

t \equiv - \overline{f'(r)}(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] Generalizations

Suppose A is a commutative ring, complete with respect to an ideal m ⊂ A, and let f(x) ∈ 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) ≡ 0 mod f ′(a)²m

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

f(b) = 0

and

b ≡ a mod f ′(a)m.

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

This result has been generalized to several variables by Nicholas 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] References

  • Eisenbud, David. Commutative Algebra with a View Toward Algebraic Geometry. Springer Graduate Texts in Mathematics, no. 150.
  • Milne, J. S. Étale Cohomology. Princeton, 1980.
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