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| In [[number theory]], '''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|''p''-adic]] absolute value.<ref>{{cite book |last=Koblitz |first=Neal |authorlink=Neal Koblitz |title=P-adic numbers, p-adic analysis, and zeta-functions |year=1984 |publisher=Springer-Verlag |location=New York |isbn=978-0-387-96017-3 |url=http://www.springer.com/mathematics/numbers/book/978-0-387-96017-3 |edition=2nd |accessdate=24 August 2012 |page=3 |quote='''Theorem 1''' (Ostrowski). ''Every nontrivial norm ‖ ‖ on ℚ is equivalent to | |<sub>p</sub> for some prime p or for p = ∞.}}</ref>
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| == Definitions ==
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| Two [[absolute value (algebra)|absolute value]]s <math>|\cdot|</math> and <math>|\cdot|_{\ast}</math> on a [[field (mathematics)|field]] '''K''' are defined to be '''equivalent''' if there exists a real number ''c'' > 0 such that
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| :<math>|x|_{\ast} = |x|^{c} \text{ for all } x \in \mathbf{K}.</math>
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| The '''trivial absolute value''' on any field '''K''' is defined to be
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| :<math>|x|_{0} := \begin{cases} 0, & \text{if } x = 0 \\ 1, & \text{if } x \ne 0. \end{cases} </math>
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| The '''real absolute value''' on the [[rational numbers|rationals]] '''Q''' is the normal absolute value on the [[real numbers|reals]], defined to be
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| :<math>|x|_\infty := \begin{cases} x, & \text{if } x \ge 0 \\ -x, & \text{if } x < 0. \end{cases} </math>
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| This is sometimes written with a subscript 1 instead of infinity.
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| For a [[prime number]] ''p'', the '''''p''-adic absolute value''' on '''Q''' is defined as follows: any non-zero rational ''x'', can be written uniquely as <math>x=p^{n}\dfrac{a}{b}</math> with ''a'', ''b'' and ''p'' [[pairwise coprime]] and <math>n\in\mathbf{Z}</math> some integer; so we define
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| :<math>|x|_{p} := \begin{cases} 0, & \text{if } x = 0 \\ p^{-n}, & \text{if } x \ne 0. \end{cases} </math>
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| == Proof ==
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| {{unreferenced section|date=June 2013}}
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| Consider a non-trivial absolute value on the rationals <math>(\mathbf{Q},|\cdot|_{\ast})</math>. We consider two cases, (i) <math>\exists{n\in\mathbf{N}},|n|_{\ast}>1</math> and (ii) <math>\forall{n\in\mathbf{N}},|n|_{\ast}\leq 1</math>. It suffices for us to consider the valuation of integers greater than one. For if we find some <math>c\in\mathbf{R}^{+}</math> for which <math>|n|_\ast=|n|^c_{\ast\ast}</math> for all naturals greater than one; then this relation trivially holds for 0 and 1, and for positive rationals
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| <math>|m/n|_\ast
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| =|m|_\ast/|n|_\ast
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| =|m|^c_{\ast\ast}/|n|^c_{\ast\ast}
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| =(|m|_{\ast\ast}/|n|_{\ast\ast})^c
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| =|m/n|^c_{\ast\ast}</math>;
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| and for negative rationals <math>|{-}x|_\ast=|x|_\ast=|x|^c_\infty=|{-}x|^c_\infty</math>.
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| === Case I: ∃''n'' ∈ '''N''' |''n''| > 1 ===
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| Consider the following calculation. Let <math>a,b\in\mathbf{N},>1</math>. Let <math>n\in\mathbf{N},>0</math>. Expressing ''b''<sup>''n''</sup> in [[radix|base]] ''a'' yields <math>b^n=\Sigma_{i{<}m}c_i a^i</math>, where each <math>c_i \in \{0,1,\ldots,a-1\}</math> and <math>m\leq n\log b/\log a+1</math>. Then we see, by the properties of an absolute value:
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| :<math> \begin{align} | |
| |b|_\ast^n = |b^{n}|_{\ast} &\leq am\max\{|a|_\ast^m,1\}\\
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| &\leq a(n\log_a b+1)\max\{|a|_\ast^{n\log_a b},1\}\\
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| \Rightarrow |b|_{\ast} &\leq \underbrace{\big(a(n\log_a b+1)\big)^{\frac{1}{n}}}_{\to 1\text{ as }n\to\infty} \max\{|a|_\ast^{\log_a b},1\}\\
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| \Rightarrow |b|_{\ast} &\leq \max\{|a|_\ast^{\log_{a}b},1\}.
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| \end{align}</math>
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| Now choose <math>b\in\mathbf{N},>1</math> such that <math>|b|_\ast>1</math>.Using this in the above ensures that <math>|a|_{\ast}>1</math> regardless of the choice of ''a'' (else <math>|a|_\ast^{\log_a b}\leq1</math> implying <math>|b|_\ast\leq 1</math>). Thus for any choice of ''a'', ''b'' > 1 above, we get <math>|b|_{\ast}\leq|a|_{\ast}^{\log b/\log a}</math>, i.e. <math>\log|b|_{\ast}/\log b\leq\log|a|_{\ast}/\log a </math>. By symmetry, this inequality is an equality.
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| Since ''a'', ''b'' were arbitrary, there is a constant, <math>\lambda\in\mathbf{R}^{+}</math> for which <math>\log|n|_{\ast}=\lambda\log n</math>,
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| i.e. <math>|n|_{\ast}=n^\lambda=|n|_\infty^\lambda</math> for all naturals ''n'' > 1. As per the above remarks, we easily see that for all rationals, <math>|x|_\ast=|x|_\infty^\lambda</math>, thus demonstrating equivalence to the real absolute value.
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| === Case II: ∀''n'' ∈ '''N''' |''n''| ≤ 1 ===
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| As this valuation is non-trivial, there must be a natural number for which <math>|n|_{\ast}<1</math>. Factorising this natural, <math>n = \Pi_{i<r}p_{i}^{e_{i}}</math> yields <math>|p|_{\ast}</math> must be less than 1, for at least one of the [[prime number|prime]] factors ''p'' = ''p''<sub>''j''</sub>. We claim than in fact, that this is so for ''only'' one.
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| Suppose ''per contra'' that <math>p,q</math> are distinct primes with absolute value less than 1. First, let <math>e\in\mathbf{N}^{+}</math> be such that <math>|p|_{\ast}^{e},|q|_{\ast}^{e}<1/2</math>. By the Euclidean algorithm, let <math>m,n\in\mathbf{Z}</math> be integers for which <math>mp^{e}+nq^{e}=1</math>. This yields <math>1=|1|_{\ast}\leq |m|_{\ast}|p|_\ast^{e}+|n|_{\ast}|q|_{\ast}^{e}<\frac{|m|_{\ast}+|n|_{\ast}}{2}\leq 1</math>, a contradiction.
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| So must have <math>|p|_{\ast}=\alpha<1</math> for some prime, and <math>|q|_\ast=1</math> all other primes. Letting <math>c=-\log\alpha/\log p</math>, we see that for general positive naturals <math>n=\Pi_{i<r}p_i^{e_i}</math>; <math>|n|_\ast=\Pi_{i<r}|p_i|_\ast^{e_i}=|p_{j}|_\ast^{e_j}=(p^{-e_j})^c=|n|_p^c</math>. As per the above remarks we see that <math>|x|_{\ast}=|x|_p^c</math> all rationals, implying the absolute value is equivalent to the ''p''-adic one.
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| {{NumBlk|1=|2=|3=<math>\blacksquare</math>|RawN=.}}
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| One can also show a stronger conclusion, namely that <math>|\cdot|_{\ast}:\mathbf{Q}\to\mathbf{R}</math> is a nontrivial absolute value if and only if either <math>|\cdot|_\ast=|\cdot|_\infty ^c</math> for some <math>c\in (0,1]</math> or <math>|\cdot|_\ast=|\cdot|_p^c</math> for some <math>c\in(0,\infty),p\in\mathbf{P}</math>.
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| == Another Ostrowski's theorem ==
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| Another theorem states that any field, complete with respect to an [[absolute value (algebra)#Types of absolute value|archimedean absolute value]], is (algebraically and topologically) isomorphic to either the [[real numbers]] or the [[complex numbers]]. This is sometimes also referred to as '''Ostrowski's theorem'''.<ref>Cassels (1986) p. 33</ref>
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| == See also ==
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| * [[Valuation (algebra)]]
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| * [[Absolute value]] in general
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| == References ==
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| {{reflist}}
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| * {{cite book | last=Cassels | first=J. W. S. | authorlink=J. W. S. Cassels
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| | title=Local Fields
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| | series=London Mathematical Society Student Texts
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| | volume=3
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| | publisher=[[Cambridge University Press]]
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| | year=1986
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| | isbn=0-521-31525-5
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| | zbl=0595.12006 }}
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| *{{cite book | last=Janusz | first=Gerald J.
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| | title = Algebraic Number Fields
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| | edition = 2nd
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| | publisher = American Mathematical Society
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| | year = 1996, 1997
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| | isbn = 0-8218-0429-4}}
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| *{{cite book | last=Jacobson | first=Nathan | authorlink = Nathan Jacobson
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| | title = Basic algebra II
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| | edition = 2nd
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| | year = 1989
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| | publisher = W H Freeman
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| | isbn = 0-7167-1933-9}}
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| *{{cite journal | last=Ostrowski | first=Alexander | authorlink = Alexander Ostrowski
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| | title = Über einige Lösungen der Funktionalgleichung φ(x)·φ(y)=φ(xy)
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| | edition = 2nd
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| | year = 1916
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| | journal = Acta Mathematica
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| | issn = 0001-5962
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| | volume = 41
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| | issue = 1
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| | pages = 271–284
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| | url = http://www.springerlink.com/content/96042g7576003r71/
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| | doi = 10.1007/BF02422947}}
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| [[Category:Theorems in number theory]]
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Irwin Butts is what my spouse loves to contact me although I don't truly like becoming called like that. Minnesota has always been his home but his wife wants them to transfer. Doing ceramics is what my family and I appreciate. He used to be unemployed but now he is a meter reader.
Feel free to visit my web-site :: http://srgame.co.kr/qna/21862