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| {{DISPLAYTITLE:''p''-adic number}}
| | I'm Bernadette and I live in Nieuw-Namen. <br>I'm interested in Dramatic Literature and History, Meteorology and Chinese art. I like to travel and watching NCIS.<br><br>Also visit my website - [http://it.wikipedia.org/wiki/Arthur_Falcone Arthur Falcone] |
| [[Image:3-adic integers with dual colorings.svg|thumb|The 3-adic integers, with selected corresponding characters on their [[Pontryagin dual]] group]]
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| In [[mathematics]] the '''{{mvar|p}}-adic number system''' for any [[prime number]] {{mvar|p}} extends the ordinary [[arithmetic]] of the [[rational numbers]] in a way different from the extension of the rational [[number system]] to the [[real number|real]] and [[complex number]] systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or [[absolute value]]. In particular, {{mvar|p}}-adic numbers have the interesting property that they are said to be close when their difference is divisible by a high power of {{mvar|p}} – the higher the power the closer they are. This property enables {{mvar|p}}-adic numbers to encode [[Modular arithmetic|congruence]] information in a way that turns out to have powerful applications in [[number theory]] including, for example, in [[Wiles's proof of Fermat's Last Theorem|the famous proof]] of [[Fermat's Last Theorem]] by [[Andrew Wiles]].<ref>F. Q. Gouvêa, A Marvelous Proof, The American Mathematical Monthly, Vol. 101, No. 3 (Mar., 1994), pp. 203–222</ref>
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| {{mvar|p}}-adic numbers were first described by [[Kurt Hensel]] in 1897,<ref>{{cite journal | last = Hensel | first = Kurt | title = Über eine neue Begründung der Theorie der algebraischen Zahlen | journal = [http://www.digizeitschriften.de/resolveppn/PPN37721857X&L=2 Jahresbericht der Deutschen Mathematiker-Vereinigung] | volume = 6 | year = 1897 | issue = 3 | pages = 83–88 | url = http://www.digizeitschriften.de/resolveppn/GDZPPN00211612X&L=2}}</ref> though with hindsight some of [[Ernst Kummer|Kummer's]] earlier work can be interpreted as implicitly using {{mvar|p}}-adic numbers.<ref>{{citation|title=Theory of Algebraic Functions of One Variable|volume=39|series=History of mathematics|first1=Richard|last1=Dedekind|author1-link=Richard Dedekind|first2=Heinrich|last2=Weber|author2-link=Heinrich Martin Weber|publisher=American Mathematical Society|year=2012|isbn=9780821890349}}. Translation into English of ''Theorie der algebraischen Functionen einer Veränderlichen'' (1882) by [[John Stillwell]]. Translator's introduction, [http://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."</ref> The {{mvar|p}}-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of [[power series]] methods into number theory. Their influence now extends far beyond this. For example, the field of [[p-adic analysis|{{mvar|p}}-adic analysis]] essentially provides an alternative form of [[calculus]].
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| More formally, for a given prime {{mvar|p}}, the [[field (mathematics)|field]] '''Q'''<sub>''p''</sub> of {{mvar|p}}-adic numbers is a [[complete space|completion]] of the [[rational number]]s. The field '''Q'''<sub>''p''</sub> is also given a [[topological space|topology]] derived from a [[metric space|metric]], which is itself derived from an [[p-adic order|alternative valuation]] on the rational numbers. This metric space is [[completeness (topology)|complete]] in the sense that every [[Cauchy sequence]] converges to a point in '''Q'''<sub>''p''</sub>. This is what allows the development of calculus on '''Q'''<sub>''p''</sub>, and it is the interaction of this analytic and algebraic structure which gives the {{mvar|p}}-adic number systems their power and utility.
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| The {{mvar|p}} in ''p-adic'' is a [[variable (mathematics)|variable]] and may be replaced with a constant (yielding, for instance, "the 2-adic numbers") or another ''placeholder variable'' (for expressions such as "the ℓ-adic numbers").
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| == ''p''-adic expansions ==
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| {{unreferenced section|date=June 2013}}
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| When dealing with natural numbers, if we take {{mvar|p}} to be a fixed prime number, then any positive [[integer]] can be written as a base {{mvar|p}} expansion in the form
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| :<math>\sum_{i=0}^n a_i p^i</math>
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| where the ''a''<sub>''i''</sub> are integers in {0, … , {{math|''p'' − 1}}}. For example, the [[binary numeral system|binary]] expansion of 35 is 1·2<sup>5</sup> + 0·2<sup>4</sup> + 0·2<sup>3</sup> + 0·2<sup>2</sup> + 1·2<sup>1</sup> + 1·2<sup>0</sup>, often written in the shorthand notation 100011<sub>2</sub>.
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| The familiar approach to extending this description to the larger domain of the rationals (and, ultimately, to the reals) is to use sums of the form:
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| :<math>\pm\sum_{i=-\infty}^n a_i p^i.</math>
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| A definite meaning is given to these sums based on [[Cauchy sequence]]s, using the [[absolute value]] as metric. Thus, for example, 1/3 can be expressed in base 5 as the limit of the sequence 0.1313131313...<sub>5</sub>. In this formulation, the integers are precisely those numbers for which ''a''<sub>''i''</sub> = 0 for all ''i'' < 0.
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| With ''p''-adic numbers, on the other hand, we choose to extend the base {{mvar|p}} expansions in a different way. Because in the {{mvar|p}}-adic world high positive powers of {{mvar|p}} are small and high negative powers are large, we consider infinite sums of the form:
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| :<math>\sum_{i=k}^{\infty} a_i p^i</math>
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| where ''k'' is some (not necessarily positive) integer. With this approach we obtain the '''{{mvar|p}}-adic expansions''' of the {{mvar|p}}-adic numbers. Those {{mvar|p}}-adic numbers for which ''a''<sub>''i''</sub> = 0 for all ''i'' < 0 are also called the '''{{mvar|p}}-adic integers'''.
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| As opposed to real number expansions which extend to the ''right'' as sums of ever smaller, increasingly negative powers of the base {{mvar|p}}, {{mvar|p}}-adic numbers may expand to the ''left'' forever, a property that can often be true for the {{mvar|p}}-adic integers. For example, consider the {{mvar|p}}-adic expansion of 1/3 in base 5. It can be shown to be …1313132{{sub|5}}, i.e., the [[limit of a sequence|limit of the sequence]] 2{{sub|5}}, 32{{sub|5}}, 132{{sub|5}}, 3132{{sub|5}}, 13132{{sub|5}}, 313132{{sub|5}}, 1313132{{sub|5}}, … :
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| :<math>\dfrac{5^2-1}{3}=\dfrac{44_5}{3} = 13_5; \,
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| \dfrac{5^4-1}{3}=\dfrac{4444_5}{3} = 1313_5</math>
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| :<math>\Rightarrow-\dfrac{1}{3}=\dots 1313_5</math>
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| :<math>\Rightarrow-\dfrac{2}{3}=\dots 1313_5 \times 2 = \dots 3131_5</math>
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| :<math>\Rightarrow\dfrac{1}{3} = -\dfrac{2}{3}+1 = \dots 3132_5.</math>
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| Multiplying this infinite sum by 3 in base 5 gives …0000001{{sub|5}}. As there are no negative powers of 5 in this expansion of 1/3 (i.e. no numbers to the right of the decimal point), we see that 1/3 satisfies the definition of being a {{mvar|p}}-adic integer in base 5.
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| More formally, the {{mvar|p}}-adic expansions can be used to define the [[field (mathematics)|field]] '''Q'''<sub>''p''</sub> of '''{{mvar|p}}-adic numbers''' while the {{mvar|p}}-adic integers form a [[subring]] of '''Q'''<sub>''p''</sub>, denoted '''Z'''<sub>''p''</sub>. (Not to be confused with the [[Modular arithmetic#Ring of congruence classes|ring of integers modulo {{mvar|p}}]] which is also sometimes written '''Z'''<sub>''p''</sub>. To avoid ambiguity, '''Z'''/''p'''''Z''' or '''Z'''/''(p)'' are often used to represent the integers modulo {{mvar|p}}.)
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| While it is possible to use the approach above to define {{mvar|p}}-adic numbers and explore their properties, just as in the case of real numbers other approaches are generally preferred. Hence we want to define a notion of infinite sum which makes these expressions meaningful, and this is most easily accomplished by the introduction of the {{mvar|p}}-adic metric. Two different but equivalent solutions to this problem are presented in the ''[[#Constructions|Constructions]]'' section below.
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| ==Notation==
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| There are several different conventions for writing {{mvar|p}}-adic expansions. So far this article has used a notation for {{mvar|p}}-adic expansions in which [[exponentiation|powers]] of {{mvar|p}} increase from right to left. With this right-to-left notation the 3-adic expansion of {{frac|1|5}}, for example, is written as
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| :<math>\dfrac{1}{5}=\dots 121012102_3.</math>
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| When performing arithmetic in this notation, digits are [[carry (arithmetic)|carried]] to the left. It is also possible to write {{mvar|p}}-adic expansions so that the powers of {{mvar|p}} increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of {{frac|1|5}} is
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| :<math>\dfrac{1}{5}=2.01210121\dots_3\mbox{ or }\dfrac{1}{15}=20.1210121\dots_3.</math>
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| {{mvar|p}}-adic expansions may be written with other sets of digits instead of {0, 1, …, {{math|''p'' − 1}}}. For example, the 3-adic expansion of <sup>1</sup>/<sub>5</sub> can be written using [[balanced ternary]] digits {<u>1</u>,0,1} as
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| :<math>\dfrac{1}{5}=\dots\underline{1}11\underline{11}11\underline{11}11\underline{1}_3.</math>
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| In fact any set of {{mvar|p}} integers which are in distinct residue classes [[modular arithmetic|modulo]] {{mvar|p}} may be used as {{mvar|p}}-adic digits. In number theory, [[Witt vector|Teichmüller representatives]] are sometimes used as digits.<ref>{{citation|title=Handbook of Algebra, Volume 6|editor-first=M.|editor-last=Hazewinkel|publisher=Elsevier|year=2009|isbn=9780080932811|page=342|url=http://books.google.com/books?id=yimXZ-7L9ZoC&pg=PA342}}.</ref>
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| == Constructions ==
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| === Analytic approach ===
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| {{see also|p-adic order}}
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| {| align=right cellpadding=4px style="margin-left:2em; margin-bottom: 2ex" cellspacing=0
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| | colspan=2 bgcolor=#333333 align=center style="color:white" |{{mvar|p}} = 2
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| | colspan=8 bgcolor=#999999 align=center |<small>← distance = 1 →</small>
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| |-
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| ! rowspan=4 bgcolor=#BBFFFF |De-<br/>ci-<br/>mal
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| ! rowspan=4 bgcolor=#FFCC66 |Bi-<br/>nary
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| | colspan=4 bgcolor=#AACC66 align=center |<small>← d = ½ →</small>
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| | colspan=4 bgcolor=#CC66CC align=center |<small>← d = ½ →</small>
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| |-
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| | colspan=2 bgcolor=#88CC00 align=center style="font-size:x-small" |‹ d=¼ ›
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| | colspan=2 bgcolor=#B5CC88 align=center style="font-size:x-small" |‹ d=¼ ›
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| | colspan=2 bgcolor=#CC00CC align=center style="font-size:x-small" |‹ d=¼ ›
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| | colspan=2 bgcolor=#CC88CC align=center style="font-size:x-small" |‹ d=¼ ›
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| |-
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| | bgcolor=#AAFF00 align=center style="font-size:x-small" |‹⅛›
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| | bgcolor=#669900 align=center style="font-size:x-small; color:white" |‹⅛›
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| | bgcolor=#EAFFBF align=center style="font-size:x-small" |‹⅛›
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| | bgcolor=#789966 align=center style="font-size:x-small; color:white" |‹⅛›
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| | bgcolor=#FF00FF align=center style="font-size:x-small" |‹⅛›
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| | bgcolor=#990099 align=center style="font-size:x-small; color:white" |‹⅛›
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| | bgcolor=#FFBFFF align=center style="font-size:x-small" |‹⅛›
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| | bgcolor=#996699 align=center style="font-size:x-small; color:white" |‹⅛›
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| |-
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| | colspan=8 |................................................
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| |-
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| | align=right bgcolor=#BBFFFF |17
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| | align=right bgcolor=#FFCC66 | 10001
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| | bgcolor=#AAFF00 |
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| | bgcolor=#669900 rowspan=5 |
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| | bgcolor=#EAFFBF rowspan=7 |
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| | bgcolor=#789966 rowspan=3 |
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| | bgcolor=#FF00FF | '''J'''
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| | bgcolor=#990099 rowspan=4 |
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| | bgcolor=#FFBFFF rowspan=6 |
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| | bgcolor=#996699 rowspan=2 |
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| |-
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| | align=right bgcolor=#BBFFFF |16
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| | align=right bgcolor=#FFCC66 |10000
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| | bgcolor=#AAFF00 | '''J'''
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| | bgcolor=#FF00FF rowspan=7 |
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| |-
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| | align=right bgcolor=#BBFFFF |15
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| | align=right bgcolor=#FFCC66 |1111
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| | bgcolor=#AAFF00 rowspan=7 |
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| | bgcolor=#996699 style="color:white" | '''L'''
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| |-
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| | align=right bgcolor=#BBFFFF |14
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| | align=right bgcolor=#FFCC66 |1110
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| | bgcolor=#789966 style="color:white" | '''L'''
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| | bgcolor=#996699 rowspan=7 |
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| |-
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| | align=right bgcolor=#BBFFFF |13
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| | align=right bgcolor=#FFCC66 |1101
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| | bgcolor=#789966 rowspan=7 |
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| | bgcolor=#990099 style="color:white" | '''L'''
| |
| |-
| |
| | align=right bgcolor=#BBFFFF |12
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| | align=right bgcolor=#FFCC66 |1100
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| | bgcolor=#669900 style="color:white" | '''L'''
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| | bgcolor=#990099 rowspan=7 |
| |
| |-
| |
| | align=right bgcolor=#BBFFFF |11
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| | align=right bgcolor=#FFCC66 |1011
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| | bgcolor=#669900 rowspan=7 |
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| | bgcolor=#FFBFFF | '''L'''
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| |-
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| | align=right bgcolor=#BBFFFF |10
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| | align=right bgcolor=#FFCC66 |1010
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| | bgcolor=#EAFFBF | '''L'''
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| | bgcolor=#FFBFFF rowspan=7 |
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| |-
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| | align=right bgcolor=#BBFFFF |9
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| | align=right bgcolor=#FFCC66 |1001
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| | bgcolor=#EAFFBF rowspan=7 |
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| | bgcolor=#FF00FF | '''L'''
| |
| |-
| |
| | align=right bgcolor=#BBFFFF |8
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| | align=right bgcolor=#FFCC66 |1000
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| | bgcolor=#AAFF00 | '''L'''
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| | bgcolor=#FF00FF rowspan=7 |
| |
| |-
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| | align=right bgcolor=#BBFFFF |7
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| | align=right bgcolor=#FFCC66 |111
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| | bgcolor=#AAFF00 rowspan=7 |
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| | bgcolor=#996699 style="color:white" |'''L'''
| |
| |-
| |
| | align=right bgcolor=#BBFFFF |6
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| | align=right bgcolor=#FFCC66 |110
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| | bgcolor=#789966 style="color:white" |'''L'''
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| | bgcolor=#996699 rowspan=7 |
| |
| |-
| |
| | align=right bgcolor=#BBFFFF |5
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| | align=right bgcolor=#FFCC66 |101
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| | bgcolor=#789966 rowspan=7 |
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| | bgcolor=#990099 style="color:white" |'''L'''
| |
| |-
| |
| | align=right bgcolor=#BBFFFF |4
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| | align=right bgcolor=#FFCC66 |100
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| | bgcolor=#669900 style="color:white" |'''L'''
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| | bgcolor=#990099 rowspan=7 |
| |
| |-
| |
| | align=right bgcolor=#BBFFFF |3
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| | align=right bgcolor=#FFCC66 |11
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| | bgcolor=#669900 rowspan=7 |
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| | bgcolor=#FFBFFF |'''L'''
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| |-
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| | align=right bgcolor=#BBFFFF |2
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| | align=right bgcolor=#FFCC66 |10
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| | bgcolor=#EAFFBF |'''L'''
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| | bgcolor=#FFBFFF rowspan=7 |
| |
| |-
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| | align=right bgcolor=#BBFFFF |1
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| | align=right bgcolor=#FFCC66 |1
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| | bgcolor=#EAFFBF rowspan=6 |
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| | bgcolor=#FF00FF |'''L'''
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| |-
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| | align=right bgcolor=#BBFFFF |0
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| | align=right bgcolor=#FFCC66 |0…000
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| | bgcolor=#AAFF00 |'''L'''
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| | bgcolor=#FF00FF rowspan=5 |
| |
| |-
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| | align=right bgcolor=#BBFFFF |−1
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| | align=right bgcolor=#FFCC66 |[[two's complement|1…111]]
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| | bgcolor=#AAFF00 rowspan=4 |
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| | bgcolor=#996699 style="color:white" | '''J'''
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| |-
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| | align=right bgcolor=#BBFFFF |−2
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| | align=right bgcolor=#FFCC66 |1…110
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| | bgcolor=#789966 style="color:white" | '''J'''
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| | bgcolor=#996699 rowspan=3 |
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| |-
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| | align=right bgcolor=#BBFFFF |−3
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| | align=right bgcolor=#FFCC66 |1…101
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| | bgcolor=#789966 rowspan=2 |
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| | bgcolor=#990099 style="color:white" | '''J'''
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| |-
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| | align=right bgcolor=#BBFFFF |−4
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| | align=right bgcolor=#FFCC66 |1…100
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| | bgcolor=#669900 style="color:white" | '''J'''
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| | bgcolor=#990099 |
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| |-
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| ! bgcolor=#BBFFFF |Dec
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| ! bgcolor=#FFCC66 |Bin
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| | colspan=8 |················································
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| |-
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| | bgcolor=#BBFFFF | <br/> <br/> <br/> <br/> <br/> <br/> <br/> <br/> <br/>
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| | colspan=9 style="font-size:small" width=32em |2-adic ({{math|1= {{mvar|p}} = [[2 (number)|2]] }}) arrangement of integers, from left to right. This shows a hierarchical subdivision pattern common for [[ultrametric space]]s. Points within a distance 1/8 are grouped in one colored strip. A pair of strips within a distance 1/4 has the same [[chromaticity|chroma]], four strips within a distance 1/2 have the same [[hue]]. The hue is determined by the [[least significant bit]], the [[colorfulness|saturation]] – by the next (2<sup>1</sub>) bit, and the [[brightness]] depends on the value of 2<sup>2</sub> bit. Bits (digit places) which are less significant for the usual metric are more significant for the {{mvar|p}}-adic distance.
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| |}
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| [[Image:3-adic metric on integers.svg|thumb|right|256px|Similar picture for {{mvar|p}} = [[3 (number)|3]] (click to enlarge) shows three closed balls of radius 1/3, where each consists of 3 balls of radius 1/9]]
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| The [[real number]]s can be defined as [[equivalence class]]es of [[Cauchy sequence]]s of [[rational number]]s; this allows us to, for example, write 1 as 1.000… = [[0.999…]] . The definition of a Cauchy sequence relies on the [[metric space|metric]] chosen, though, so if we choose a different one, we can construct numbers other than the real numbers. The usual metric which yields the real numbers is called the [[Euclidean metric]].
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| For a given prime {{mvar|p}}, we define the ''p-adic absolute value'' in '''Q''' as follows:
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| for any non-zero rational number {{mvar|x}}, there is a unique integer {{mvar|n}} allowing us to write {{math|1=''x'' = ''p''<sup>''n''</sup>(''a''/''b'')}}, where neither of the integers ''a'' and ''b'' is [[divisor|divisible]] by {{mvar|p}}. Unless the numerator or denominator of {{mvar|x}} in lowest terms contains {{mvar|p}} as a factor, {{mvar|n}} will be 0. Now define {{math|1=|''x''|<sub>''p''</sub> = ''p''<sup>−''n''</sup>}}. We also define {{math|1=|0|<sub>''p''</sub> = 0}}.
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| For example with {{math|1=''x'' = 63/550 = 2<sup>−1</sup>·3<sup>2</sup>·5<sup>−2</sup>·7·11<sup>−1</sup>}}<br>
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| :<math>
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| \begin{align}
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| |x|_2 & = 2 \\[6pt]
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| |x|_3 & = 1/9 \\[6pt]
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| |x|_5 & = 25 \\[6pt]
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| |x|_7 & = 1/7 \\[6pt]
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| |x|_{11} & = 11 \\[6pt]
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| |x|_{\text{any other prime}} & = 1.
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| \end{align}
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| </math>
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| This definition of {{math||''x''|<sub>''p''</sub>}} has the effect that high powers of {{mvar|p}} become "small".
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| By the [[fundamental theorem of arithmetic]], for a given non-zero rational number ''x'' there is a unique finite set of distinct primes <math>p_1, \ldots, p_r</math> and a corresponding sequence of non-zero integers <math>a_1, \ldots, a_r</math> such that:
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| :<math> |x| = p_1^{a_1}\ldots p_r^{a_r}.</math>
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| It then follows that <math> |x|_{p_i} = p_i^{-a_i} </math> for all <math> 1\leq i\leq r </math>, and <math>|x|_p = 1\,</math> for any other prime <math> p \notin \{p_1,\ldots, p_r\}.</math>
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| It is a [[Ostrowski's theorem|theorem of Ostrowski]] that each [[absolute value (algebra)|absolute value]] on '''Q''' is equivalent either to the Euclidean absolute value, the [[absolute value (algebra)#Types of absolute value|trivial absolute value]], or to one of the {{mvar|p}}-adic absolute values for some prime {{mvar|p}}. So the only norms on '''Q''' modulo equivalence are the absolute value, the trivial absolute value and the {{mvar|p}}-adic absolute value which means that there are only as many completions (with respect to a norm) of '''Q'''.
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| The {{mvar|p}}-adic absolute value defines a metric d<sub>''p''</sub> on '''Q''' by setting
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| :<math>d_p(x,y)=|x-y|_p \,\!</math>
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| The field '''Q'''<sub>''p''</sub> of {{mvar|p}}-adic numbers can then be defined as the [[completion (topology)|completion]] of the metric space ('''Q''', d<sub>''p''</sub>); its elements are equivalence classes of Cauchy sequences, where two sequences are called equivalent if their difference converges to zero. In this way, we obtain a complete metric space which is also a field and contains '''Q'''.
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| It can be shown that in '''Q'''<sub>''p''</sub>, every element ''x'' may be written in a unique way as
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| :<math>\sum_{i=k}^{\infty} a_i p^i</math>
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| where ''k'' is some integer such that ''a''<sub>''k''</sub> ≠ ''0'' and each ''a''<sub>''i''</sub> is in {0, …, {{math|''p'' − 1}} }. This series [[limit of a sequence|converges]] to ''x'' with respect to the metric d<sub>''p''</sub>.
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| With this absolute value, the field '''Q'''<sub>''p''</sub> is a [[local field]].
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| === Algebraic approach ===
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| In the algebraic approach, we first define the ring of {{mvar|p}}-adic integers, and then construct the field of fractions of this ring to get the field of {{mvar|p}}-adic numbers.
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| We start with the [[inverse limit]] of the rings
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| '''Z'''/''p''<sup>''n''</sup>'''Z''' (see [[modular arithmetic]]): a {{mvar|p}}-adic integer is then a sequence
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| (''a<sub>n</sub>'')<sub>''n''≥1</sub> such that ''a''<sub>''n''</sub> is in
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| '''Z'''/''p''<sup>''n''</sup>'''Z''', and if ''n'' ≤ ''m'', then
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| ''a<sub>n</sub>'' ≡ ''a''<sub>''m''</sub> (mod ''p''<sup>''n''</sup>).
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| Every natural number ''m'' defines such a sequence (''a''<sub>''n''</sub>) by ''a''<sub>''n''</sub> = ''m'' mod ''p<sup>n</sup>'' and can therefore be regarded as a {{mvar|p}}-adic integer. For example, in this case 35 as a 2-adic integer would be written as the sequence (1, 3, 3, 3, 3, 35, 35, 35, …).
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| The operators of the ring amount to pointwise addition and multiplication of such sequences. This is well defined because addition and multiplication commute with the "mod" operator, see [[modular arithmetic]].
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| Moreover, every sequence (''a''<sub>''n''</sub>) where the first element is not 0 has an inverse. In that case, for every ''n'', ''a<sub>n</sub>'' and ''p'' are [[coprime]], and so ''a<sub>n</sub>'' and ''p<sup>n</sup>'' are relatively prime. Therefore, each ''a<sub>n</sub>'' has an inverse mod ''p<sup>n</sup>'', and the sequence of these inverses, (''b<sub>n</sub>''), is the sought inverse of (''a<sub>n</sub>''). For example, consider the {{mvar|p}}-adic integer corresponding to the natural number 7; as a 2-adic number, it would be written (1, 3, 7, 7, 7, 7, 7, ...). This object's inverse would be written as an ever-increasing sequence that begins (1, 3, 7, 7, 23, 55, 55, 183, 439, 439, 1463 ...). Naturally, this 2-adic integer has no corresponding natural number.
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| Every such sequence can alternatively be written as a [[series (mathematics)|series]]. For instance, in the 3-adics, the sequence (2, 8, 8, 35, 35, ...) can be written as {{math|2 + 2·3 + 0·3<sup>2</sup> + 1·3<sup>3</sup> + 0·3<sup>4</sup> + ...}} The [[partial sum]]s of this latter series are the elements of the given sequence.
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| The ring of {{mvar|p}}-adic integers has no zero divisors, so we can take the [[field of fractions]] to get the field '''Q'''<sub>''p''</sub> of {{mvar|p}}-adic numbers. Note that in this field of fractions, every non-integer {{mvar|p}}-adic number can be uniquely written as {{math|''p''<sup>−''n''</sup>'' u''}} with a [[natural number]] ''n'' and a unit in the {{mvar|p}}-adic integers ''u''. This means that
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| :<math> \mathbf{Q}_p=\operatorname{Quot}\left(\mathbf{Z}_p\right)\cong (p^{\mathbf{N}})^{-1}\mathbf{Z}_p.</math>
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| Note that {{math|''S''<sup>−1</sup> ''A''}}<!-- don't use <math> here until a space after unary minus will not be removed in HTML output -->, where <math>S=p^{\mathbf{N}}=\{p^{n}:n\in\mathbf{N}\}</math> is a multiplicative subset (contains the unit and closed under multiplication) of a commutative ring with unit <math>A</math>, is an algebraic construction called the '''ring of fractions''' of <math>A</math> by <math>S</math>.
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| == Properties ==
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| The ring of {{mvar|p}}-adic integers is the [[inverse limit]] of the finite rings '''Z'''/''p''<sup>''k''</sup>'''Z''', but is nonetheless [[uncountable set|uncountable]],<ref>Robert (2000) Section 1.1</ref> and has the [[cardinality of the continuum]]. Accordingly, the field '''Q'''<sub>''p''</sub> is uncountable. The [[endomorphism ring]] of the [[Prüfer group|Prüfer {{mvar|p}}-group]] of rank ''n'', denoted '''Z'''(''p''<sup>∞</sup>)<sup>''n''</sup>, is the ring of ''n''×''n'' matrices over the {{mvar|p}}-adic integers; this is sometimes referred to as the [[Tate module]].
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| The {{mvar|p}}-adic numbers contain the rational numbers '''Q''' and form a field of [[characteristic (algebra)|characteristic]] 0. This field cannot be turned into an [[ordered field]].
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| Let the [[topology (structure)|topology]] τ on '''Z'''<sub>p</sub> be defined by taking as a [[basis (topology)|basis]] all sets of the form {{math|U<sub>''a''</sub>(''n'') {{=}}}} { {{math|''n'' + λ p<sup>''a''</sup>}} for λ in '''Z'''<sub>''p''</sub> and ''a'' in '''N'''}. Then '''Z'''<sub>p</sub> is a [[compactification (mathematics)|compactification]] of '''Z''', under the derived topology (it is ''not'' a compactification of '''Z''' with its usual discrete topology). The [[relative topology]] on '''Z''' as a subset of '''Z'''<sub>''p''</sub> is called the [[p-adic topology|{{mvar|p}}-adic topology]] on '''Z'''.
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| The topology of the set of {{mvar|p}}-adic integers is that of a [[Cantor set]]; the topology of the set of {{mvar|p}}-adic numbers is that of a Cantor set minus a point (which would naturally be called infinity).<ref>Robert (2000) Section 2.3</ref> In particular, the space of {{mvar|p}}-adic integers is [[Compact space|compact]] while the space of {{mvar|p}}-adic numbers is not; it is only [[locally compact]]. As [[metric space]]s, both the {{mvar|p}}-adic integers and the {{mvar|p}}-adic numbers are [[completeness (topology)|complete]].<ref>Gouvêa (2000) Corollary 3.3.8</ref>
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| The real numbers have only a single proper [[algebraic extension]], the [[complex number]]s; in other words, this quadratic extension is already [[algebraically closed field|algebraically closed]]. By contrast, the [[algebraic closure]] of the {{mvar|p}}-adic numbers has infinite degree,<ref>Gouvêa (2000) Corollary 5.3.10</ref> i.e. '''Q'''<sub>''p''</sub> has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the {{mvar|p}}-adic valuation to the algebraic closure of '''Q'''<sub>''p''</sub>, it is not (metrically) complete.<ref>Gouvêa (2000) Theorem 5.7.4</ref><ref name=C149>Cassels (1986) p.149</ref> Its (metric) completion is called '''C'''<sub>''p''</sub> or Ω<sub>''p''</sub>.<ref name=C149/><ref name=K13>Koblitz (1980) p.13</ref> Here an end is reached, as '''C'''<sub>''p''</sub> is algebraically closed.<ref name=C149/><ref>Gouvêa (2000) Proposition 5.7.8</ref> Unlike the complex field, '''C'''<sub>''p''</sub> is not locally compact.<ref name=K13/>
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| The field '''C'''<sub>''p''</sub> is algebraically isomorphic to the field '''C''' of complex numbers, so we may regard '''C'''<sub>''p''</sub> as the complex numbers endowed with an exotic metric. It should be noted that the proof of existence of such a field isomorphism relies on the [[axiom of choice]], and does not provide an explicit example of such an isomorphism.
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| The {{mvar|p}}-adic numbers contain the {{mvar|n}}th [[cyclotomic field]] ({{math|n > 2}}) if and only if {{mvar|n}} divides {{math|''p'' − 1}}.<ref>Gouvêa (2000) Proposition 3.4.2</ref> For instance, the {{mvar|n}}th cyclotomic field is a subfield of '''Q'''<sub>13</sub> if and only if ''n'' = 1, 2, 3, 4, 6, or 12. In particular, there is no multiplicative {{mvar|p}}-[[torsion (algebra)|torsion]] in the {{mvar|p}}-adic numbers, if {{math|''p'' > 2}}. Also, −1 is the only non-trivial torsion element in 2-adic numbers.
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| Given a natural number ''k'', the index of the multiplicative group of the ''k''th powers of the non-zero elements of '''Q'''<sub>''p''</sub> in the multiplicative group of '''Q'''<sub>''p''</sub> is finite.
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| The number ''[[e (mathematical constant)|e]]'', defined as the sum of reciprocals of [[factorial]]s, is not a member of any {{mvar|p}}-adic field; but {{math|''e''<sup>''p''</sup>}} is a {{mvar|p}}-adic number for all {{mvar|p}} except 2, for which one must take at least the fourth power.<ref>Robert (2000) Section 4.1</ref> (Thus a number with similar properties as {{mvar|e}} – namely a {{mvar|p}}th root of {{math|''e''<sup>''p''</sup>}} – is a member of the algebraic closure of the {{mvar|p}}-adic numbers for all {{mvar|p}}.)
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| For reals, the only functions whose [[derivative]] is zero are the constant functions. This is not true over '''Q'''<sub>''p''</sub>.<ref>Robert (2000) Section 5.1</ref> For instance, the function
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| :''f'': '''Q'''<sub>''p''</sub> → '''Q'''<sub>''p''</sub>, ''f''(''x'') = (1/|''x''|<sub>''p''</sub>)<sup>2</sup> for ''x'' ≠ 0, ''f''(0) = 0,
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| has zero derivative everywhere but is not even [[locally constant function|locally constant]] at 0.
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| Given any elements ''r''<sub>∞</sub>, ''r''<sub>2</sub>, ''r''<sub>3</sub>, ''r''<sub>5</sub>, ''r''<sub>7</sub>, ... where ''r''<sub>''p''</sub> is in '''Q'''<sub>''p''</sub> (and '''Q'''<sub>∞</sub> stands for '''R'''), it is possible to find a sequence (''x''<sub>''n''</sub>) in '''Q''' such that for all ''p'' (including ∞), the limit of ''x''<sub>''n''</sub> in '''Q'''<sub>''p''</sub> is ''r''<sub>''p''</sub>.
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| The field '''Q'''<sub>''p''</sub> is a locally compact [[Hausdorff space]].
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| If <math>\mathbf{K}</math> is a finite [[Galois extension]] of <math>\mathbf{Q}_{p}</math>,
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| the [[Galois group]] <math>\text{Gal}(\mathbf{K}/\mathbf{Q}_{p})</math> is [[solvable group|solvable]].
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| Thus, the Galois group <math>\text{Gal}(\overline{\mathbf{Q}}_{p}/\mathbf{Q}_{p})</math> is [[prosolvable]].
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| ==Rational arithmetic==
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| [[Eric Hehner]] and [[Nigel Horspool]] proposed in 1979 the use of a {{mvar|p}}-adic representation for rational numbers on computers<ref>Eric C. R. Hehner, R. Nigel Horspool, A new representation of the rational numbers for fast easy arithmetic. SIAM Journal on Computing 8, 124–134. 1979.</ref> called [[Quote notation]]. The primary advantage of such a representation is that addition, subtraction, and multiplication can be done in a straightforward manner analogous to similar methods for binary integers; and division is even simpler, resembling multiplication. However, it has the disadvantage that representations can be much larger than simply storing the numerator and denominator in binary; for example, if {{math|2<sup>''n''</sup> − 1}} is a [[Mersenne prime]], its reciprocal will require {{math|2<sup>''n''</sup> − 1}} bits to represent.
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| == Generalizations and related concepts ==
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| The reals and the {{mvar|p}}-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general [[algebraic number field]]s, in an analogous way. This will be described now.
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| Suppose ''D'' is a [[Dedekind domain]] and ''E'' is its [[field of fractions]]. Pick a non-zero [[prime ideal]] ''P'' of ''D''. If ''x'' is a non-zero element of ''E'', then ''xD'' is a [[fractional ideal]] and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals of ''D''. We write ord<sub>''P''</sub>(''x'') for the exponent of ''P'' in this factorization, and for any choice of number ''c'' greater than 1 we can set
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| :<math>|x|_P = c^{-\operatorname{ord}_P(x)}.</math>
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| Completing with respect to this absolute value |.|<sub>''P''</sub> yields a field ''E''<sub>''P''</sub>, the proper generalization of the field of ''p''-adic numbers to this setting. The choice of ''c'' does not change the completion (different choices yield the same concept of Cauchy sequence, so the same completion). It is convenient, when the [[residue field]] ''D''/''P'' is finite, to take for ''c'' the size of ''D''/''P''.
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| For example, when ''E'' is a [[number field]], [[Ostrowski's theorem]] says that every non-trivial [[absolute value (algebra)|non-Archimedean absolute value]] on ''E'' arises as some |.|<sub>''P''</sub>. The remaining non-trivial absolute values on ''E'' arise from the different embeddings of ''E'' into the real or complex numbers. (In fact, the non-Archimedean absolute values can be considered as simply the different embeddings of ''E'' into the fields '''C'''<sub>''p''</sub>, thus putting the description of all
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| the non-trivial absolute values of a number field on a common footing.)
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| Often, one needs to simultaneously keep track of all the above mentioned completions when ''E'' is a number field (or more generally a [[global field]]), which are seen as encoding "local" information. This is accomplished by [[adele ring]]s and [[idele group]]s.
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| ==Local–global principle==
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| [[Helmut Hasse]]'s [[Hasse principle|local–global principle]] is said to hold for an equation if it can be solved over the rational numbers [[if and only if]] it can be solved over the [[real number]]s and over the {{mvar|p}}-adic numbers for every prime {{mvar|p}}. This principle holds e.g. for equations given by [[quadratic form]]s, but fails for higher polynomials in several indeterminates.
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| ==See also==
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| * [[1 + 2 + 4 + 8 + ...]]
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| * [[C-minimal theory]]
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| * [[Hensel's lemma]]
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| * [[Bijective numeration|''k''-adic notation]]
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| * [[Mahler's theorem]]
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| * [[Volkenborn integral]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{cite book |last=Bachman |first=George |title=Introduction to ''p''-adic Numbers and Valuation Theory |year=1964 |publisher=Academic Press |isbn=0-12-070268-1}}
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| *{{cite book |last=Cassels |first=J. W. S. |authorlink=J. W. S. Cassels |title=Local Fields |series=London Mathematical Society Student Texts |volume=3 |publisher=[[Cambridge University Press]] |year=1986 |isbn=0-521-31525-5 |zbl=0595.12006}}
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| *{{cite book |last=Gouvêa |first=Fernando Q. |year=2000 |title=''p''-adic Numbers: An Introduction |edition=2nd |publisher=Springer |isbn=3-540-62911-4}}
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| *{{cite book |last=Koblitz |first=Neal |authorlink=Neal Koblitz |title=''p''-adic analysis: a short course on recent work |series=London Mathematical Society Lecture Note Series |volume=46 |publisher=[[Cambridge University Press]] |year=1980 |isbn=0-521-28060-5 |zbl=0439.12011}}
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| *{{cite book |last=Koblitz |first=Neal |authorlink=Neal Koblitz |year=1996 |title=''p''-adic Numbers, ''p''-adic Analysis, and Zeta-Functions |edition=2nd |publisher=Springer |isbn=0-387-96017-1}}
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| *{{cite book |last=Robert |first=Alain M. |year=2000 |title=A Course in ''p''-adic Analysis |publisher=Springer |isbn=0-387-98669-3}}
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| *{{cite book |last=Steen |first=Lynn Arthur |authorlink=Lynn Arthur Steen |year=1978 |title=[[Counterexamples in Topology]] |publisher=Dover |isbn=0-486-68735-X}}
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| ==External links==
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| *{{MathWorld|urlname=p-adicNumber|title=p-adic Number}}
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| *{{planetmath reference|id=3118|title=p-adic integers}}
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| *[http://www.encyclopediaofmath.org/index.php/P-adic_number ''p''-adic number] at [[Encyclopaedia of Mathematics|Springer On-line Encyclopaedia of Mathematics]]
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| *[http://math.stanford.edu/~conrad/248APage/handouts/algclosurecomp.pdf Completion of Algebraic Closure] – on-line lecture notes by Brian Conrad
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| *[http://www.maths.gla.ac.uk/~ajb/dvi-ps/padicnotes.pdf An Introduction to ''p''-adic Numbers and ''p''-adic Analysis] - on-line lecture notes by Andrew Baker, 2007
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| * [http://homes.esat.kuleuven.be/~fvercaut/talks/pAdic.pdf Efficient p-adic arithmetic] (slides)
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| {{Number Systems}}
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| {{DEFAULTSORT:P-Adic Number}}
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| [[Category:Field theory]]
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| [[Category:Number theory]]
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