Editing P-adic analysis (section)

===Hensel's lemma===
{{main article|Hensel's lemma}}
Hensel's lemma, also known as Hensel's lifting lemma, named after [[Kurt Hensel]], is a result in [[modular arithmetic]], stating that if a [[polynomial equation]] has a [[Multiplicity (mathematics)#Multiplicity of a root of a polynomial|simple root]] modulo a [[prime number]] {{math|''p''}}, then this root corresponds to a unique root of the same equation modulo any higher power of {{math|''p''}}, which can be found by iteratively "[[lift (mathematics)|lift]]ing" the solution modulo successive powers of {{math|''p''}}. More generally it is used as a generic name for analogues for [[completion (ring theory)|complete]] [[commutative ring]]s (including [[p-adic field|''p''-adic field]]s in particular) of the [[Newton method]] for solving equations. Since ''p''-adic analysis is in some ways simpler than [[real analysis]], there are relatively easy criteria guaranteeing a root of a polynomial.

To state the result, let <math>f(x)</math> be a [[polynomial]] with [[integer]] (or ''p''-adic integer) coefficients, and let ''m'',''k'' be positive integers such that ''m'' ≤ ''k''.  If ''r'' is an integer such that

:<math>f(r) \equiv 0 \pmod{p^k}</math> and <math>f'(r) \not\equiv 0 \pmod{p}</math>

then there exists an integer ''s'' such that

:<math>f(s) \equiv 0 \pmod{p^{k+m}}</math> and <math>r \equiv s \pmod{p^{k}}.</math>

Furthermore, this ''s'' is unique modulo ''p''<sup>''k''+m</sup>, and can be computed explicitly as

:<math>s = r + tp^k</math> where <math>t = - \frac{f(r)}{p^k} \cdot (f'(r)^{-1}).</math>