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Categories: Modular arithmetic | Theorems

Hensel's lemma

In mathematics, Hensel's lemma, named after Kurt Hensel , is a generic name for analogues for p-adic fields 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.

One version of the lemma is as follows. Let f(x) be a polynomial with integer coefficients, k is an integer not less than 2 and p is 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 $x \equiv r \pmod{p^{k-1}}$ .

See also: Hensel ring , Henselization .

Categories: Modular arithmetic | Theorems

Last updated: 05-27-2005 15:09:55

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