Editing P-adic number (section)

{{Short description|Number system extending the rational numbers}}
{{DISPLAYTITLE:''p''-adic number}}
[[Image:3-adic integers with dual colorings.svg|thumb|The 3-adic integers, with selected corresponding characters on their [[Pontryagin dual]] group]]

In [[number theory]], given a [[prime number]] {{mvar|p}},{{efn-num|In this article, unless otherwise stated, {{mvar|p}} denotes a prime number that is fixed once for all.}} the '''{{mvar|p}}-adic numbers''' form an extension of the [[rational number]]s which is distinct from the [[real number]]s, though with some similar properties; {{mvar|p}}-adic numbers can be written in a form similar to (possibly [[infinity (mathematics)|infinite]]) [[decimal representation|decimal]]s, but with digits based on a prime number {{mvar|p}} rather than ten, and extending to the left rather than to the right.

For example, comparing the expansion of the rational number <math>\tfrac15</math> in [[Ternary numeral system|base {{math|3}}]] vs. the {{math|3}}-adic expansion,
<math display="block">\begin{alignat}{3}
\tfrac15 &{}= 0.01210121\ldots \ (\text{base } 3)
         &&{}= 0\cdot 3^0 + 0\cdot 3^{-1} + 1\cdot 3^{-2} + 2\cdot 3^{-3} + \cdots \\[5mu]
\tfrac15 &{}= \dots 121012102 \ \ (\text{3-adic})
         &&{}= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0.
\end{alignat}</math>

Formally, given a prime number {{mvar|p}}, a {{mvar|p}}-adic number can be defined as a [[series (mathematics)|series]]
<math display="block">s=\sum_{i=k}^\infty a_i p^i = a_k p^k + a_{k+1} p^{k+1} + a_{k+2} p^{k+2} + \cdots</math>
where {{mvar|k}} is an [[integer]] (possibly negative), and each <math>a_i</math> is an integer such that <math>0\le a_i < p.</math> A '''{{mvar|p}}-adic integer''' is a {{mvar|p}}-adic number such that <math>k\ge 0.</math>

In general the series that represents a {{mvar|p}}-adic number is not [[convergent series|convergent]] in the usual sense, but it is convergent for the [[p-adic absolute value|{{mvar|p}}-adic absolute value]] <math>|s|_p=p^{-k},</math> where {{mvar|k}} is the least integer {{mvar|i}} such that <math>a_i\ne 0</math> (if all <math>a_i</math> are zero, one has the zero {{mvar|p}}-adic number, which has {{math|0}} as its {{mvar|p}}-adic absolute value).

Every rational number can be uniquely expressed as the sum of a series as above, with respect to the {{mvar|p}}-adic absolute value. This allows considering rational numbers as special {{mvar|p}}-adic numbers, and alternatively defining the {{mvar|p}}-adic numbers as the [[completion (metric space)|completion]] of the rational numbers for the {{mvar|p}}-adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.

{{mvar|p}}-adic numbers were first described by [[Kurt Hensel]] in 1897,<ref>{{Harv|Hensel|1897}}</ref> though, with hindsight, some of [[Ernst Kummer|Ernst Kummer's]] earlier work can be interpreted as implicitly using {{mvar|p}}-adic numbers.<ref group="note">Translator's introduction, [https://books.google.com/books?id=Qxte2mhlEOYC&pg=PA35 page 35]: "Indeed, with hindsight it becomes apparent that a [[discrete valuation]] is behind Kummer's concept of ideal numbers." {{Harv|Dedekind|Weber|2012|p=35}}</ref>