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.

1 First form
2 Example
3 Generalizations
4 Related concepts
5 See also
6 References

[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 Z_p. 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 { r_i } 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 Z_p congruent to r₁ modulo p. Conversely, if a is a complete square in Z_p 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 $x 2 - 2 = 0$ ) in the 7-adic integers. Modulo 7, we have a solution, namely 3 (we could also take 4), so $r 1 = 3$ . Hensel's lemma then allows us to find $r 2$ as follows:

f (r 1) = 3 2 - 2 = 7

f (r 1) / p 1 = 7 / 7 = 1

f'(r 1) = 2 r 1 = 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 $r 3 = 108 = 213 7$ . 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, …, a_n) ∈ Aⁿ an approximate solution to a system of polynomials f_i(x) ∈ A[x₁, …, x_n] in the sense that

f_i(a) ≡ 0 mod m

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

f_i(b) = 0

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

b_i ≡ a_i 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 A^h containing A such that A^h is Henselian with respect to mA^h. This A^h is called the Henselization of A. If A is noetherian, A^h will also be noetherian, and A^h is manifestly algebraic as it is constructed as a limit of étale neighbourhoods. This means that A^h is usually much smaller than the completion Â while still retaining the Henselian property and remaining in the same category.

[edit] See also

[edit] References

Eisenbud, David (1995), Commutative algebra, vol. 150, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR 1322960, ISBN 978-0-387-94268-1; 978-0-387-94269-8 .
Milne, J. G. (1980), Étale cohomology, Princeton University Press, ISBN 978-0-691-08238-7 .

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Hensel's lemma

From Wikipedia, the free encyclopedia

Contents

[edit] First form

[edit] Example

[edit] Generalizations

[edit] Related concepts

[edit] See also

[edit] References

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