Editing P-adic analysis (section)

==Important results==

===Ostrowski's theorem===
{{main article|Ostrowski's theorem}}
Ostrowski's theorem, due to [[Alexander Ostrowski]] (1916), states that every non-trivial [[absolute value (algebra)|absolute value]] on the [[rational number]]s '''Q''' is equivalent to either the usual real absolute value or a [[p-adic number|{{mvar|p}}-adic]] absolute value.<ref>{{cite book |last=Koblitz |first=Neal |authorlink=Neal Koblitz |title=P-adic numbers, p-adic analysis, and zeta-functions |series=Graduate Texts in Mathematics |year=1984 |volume=58 |publisher=Springer-Verlag |location=New York |isbn=978-0-387-96017-3 |doi=10.1007/978-1-4612-1112-9 |edition=2nd  |page=3 |quote='''Theorem 1''' (Ostrowski). Every nontrivial norm ‖ ‖ on <math>\mathbb{Q}</math> is equivalent to {{math|{{abs|&nbsp;}}<sub>''p''</sub>}} for some prime {{mvar|p}} or for {{math|1=''p'' = ∞}}.}}</ref>

===Mahler's theorem===
{{main article|Mahler's theorem}}
'''Mahler's theorem''', introduced by [[Kurt Mahler]],<ref>{{Citation | last1=Mahler | first1=K. | title=An interpolation series for continuous functions of a p-adic variable | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002177846 | mr=0095821 | year=1958 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=1958 | issue=199 | pages=23–34| doi=10.1515/crll.1958.199.23 | s2cid=199546556 }}
</ref> expresses continuous ''p''-adic functions in terms of polynomials.

In any [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic]] 0, one has the following result.  Let

:<math>(\Delta f)(x)=f(x+1)-f(x)</math>

be the forward [[difference operator]].  Then for [[polynomial function]]s ''f'' we have the [[Newton series]]:

:<math>f(x)=\sum_{k=0}^\infty (\Delta^k f)(0){x \choose k},</math>

where

:<math>{x \choose k}=\frac{x(x-1)(x-2)\cdots(x-k+1)}{k!}</math>

is the ''k''th binomial coefficient polynomial.

Over the field of real numbers, the assumption that the function ''f'' is a polynomial can be weakened, but it cannot be weakened all the way down to mere [[continuous function|continuity]].

Mahler proved the following result:

'''Mahler's theorem''': If ''f'' is a continuous [[p-adic number|''p''-adic]]-valued function on the ''p''-adic integers then the same identity holds.

===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>