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...ber]]s, and [[complete valued field]]s (such as the [[p-adic numbers|''p''-adic number]]s). ===Real and complex numbers=== ...

...he '''Volkenborn integral''' is a method of [[integral|integration]] for p-adic functions. ...ion from the [[P-adic number|p-adic]] integers taking values in the p-adic numbers. The Volkenborn integral is defined by the limit, if it exists: ...

...rfect field|perfect]] of [[characteristic (algebra)|characteristic]] <math>p</math>, there is a unique multiplicative [[Section (category theory)|sectio ...mega(x)^p = \omega(x)</math> that is congruent to <math>x</math> mod <math>p</math>. It can also be defined by ...

...escription|Result in number theory showing congruences involving Bernoulli numbers}} ...nces''' are some [[Congruence relation|congruence]]s involving [[Bernoulli numbers]], found by {{harvs|txt|authorlink=Ernst Eduard Kummer|first=Ernst Eduard|l ...

{{DISPLAYTITLE:''p''-adically closed field}} ...joys a closure property that is a close analogue for [[p-adic number|''p''-adic fields]] to what [[Real closed field|real closure]] is to the [[Real number ...

==<span id="padic"></span>''p''-adic cyclotomic character== Fix {{math|''p''}} a [[prime number|prime]], and let {{math|''G''_''Q''}} denote ...

{{DISPLAYTITLE:''p''-adic exponential function}} ...]]. As in the complex case, it has an inverse function, named the '''''p''-adic logarithm'''. ...

...Hodge theory]], [[Galois representation]]s, and [[p-adic cohomology|''p''-adic cohomology]]. She studies the cohomology of <math>p</math>-adic varieties. Her contributions include: ...

{{DISPLAYTITLE:''p''-adic gamma function}} ...|Overholtzer|1952}} had previously given a definition of a different ''p''-adic analogue of the gamma function, but his function does not have satisfactory ...

{{DISPLAYTITLE:''p''-adic ''L''-function}} ...the image could be the [[p-adic number|''p''-adic numbers]] '''Q'''_''p'' or its [[algebraic closure]]. ...

===Case of ''p''-adic representations=== ...K</sub>'' (i.e. the [[Tate module]] of μ). A [[Galois representation|''p''-adic representation]] of ''G_K'' is a [[continuous map|continuous]] [[ ...

In [[number theory]], more specifically in [[p-adic analysis|''p''-adic analysis]], '''Krasner's lemma''' is a basic result relating the [[topologi ...8.1.6 of {{harvnb|Neukirch|Schmidt|Wingberg|2008}}</ref><ref>Lorenz (2008) p.78</ref> ...

...''p''</sub>, and if ''v_p'' denotes the [[p-adic valuation|''p''-adic valuation]], then :<math>\chi(G_K,M)=p^{-[K:\mathbf{Q}_p]v_p(m)}</math> ...

...a Belgian mathematician known as a pioneer of [[p-adic analysis|{{mvar|p}}-adic functional analysis]], and particularly for her work on [[locally convex to ...De Kimpe founded the series of International Conferences on <math>p</math>-adic Functional Analysis, with cofounders Javier Martínez Maurica and José Manue ...

{{Short description|Special numbers in mathematics}} ...3 | year=2007 | chapter=Algebraic theta functions and Eisenstein-Kronecker numbers | pages=63–77| bibcode=2007arXiv0709.0640B }}</ref><ref>{{Citation | last1= ...

...allows one to construct analogues of [[algebraic number field]]s in the p-adic context. ...<ref name=":0">{{Cite book|last=Koblitz|first=Neil|title=p-adic Numbers, p-adic Analysis, and Zeta-Functions|pages=57–74}}</ref> ^{pg. 58-59}. ...

| thesis_title = p-Adic L-functions for Elliptic Curves over CM Fields ...f Mathematics}}</ref> He is known for his work in [[p-adic analysis]], [[p-adic quantum mechanics]], and [[non-additive geometry]], including the [[field w ...

...]] that divides the numerators of exactly three [[Harmonic_number|harmonic numbers]]. ...the partial harmonic sums at positions ''p-1'', ''p*(p-1)'', and ''(p-1)*(p+1)''. ...

...e field to be spherically complete.<ref>{{Cite book |last=Schneider |first=P. |title=Nonarchimedean functional analysis |date=2002 |publisher=Springer | ...e. This includes, in particular, the fields '''Q'''_''p'' of [[p-adic number]]s, and any of their finite extensions. ...

...anuary 2019) was a French mathematician. He was one of the founders of [[p-adic Hodge theory]]. He was a professor at [[Paris-Sud 11 University]] from 1988 ...de variété abélienne sur Z'', [[Inventiones Mathematicae]] vol. 81, 1985, p.&nbsp;515). He introduced the concept of [[geometric Galois representation] ...