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.

Related Topics:
Mathematics - Kurt Hensel - P-adic field - Newton method - P-adic analysis - Real analysis

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One version of the lemma 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

Related Topics:
Polynomial - Integer - Prime number - Congruence

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

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

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

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with t given by

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:t equiv - overline{f'(r)}(f(r)/p^{k-1}) pmod{p}.,

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

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then

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

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for all integers t.

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Also, if f'(r) equiv 0 pmod{p} and f(r) otequiv 0 pmod{p^k},

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then f(x) equiv 0 pmod{p^k} has no solution for any x equiv r pmod{p^{k-1}}.

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