Dual wavelet - Revision history

en>Yobot: /* Definition */WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T10:02:19Z

Definition: WP:CHECKWIKI error fixes using AWB (10093)

← Older revision		Revision as of 12:02, 5 May 2014
Line 1:		Line 1:
	~~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>~~		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.<br><br>Feel free to visit my web-site :: [http://srgame.co.kr/qna/21862 http://srgame.co.kr/qna/21862]

	~~== Definitions ==~~
	~~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~~
	~~:<math>\|x\|_{\ast} = \|x\|^{c} \text{ for all } x \in \mathbf{K}~~.~~</math>~~

	~~The '''trivial absolute value''' on any field '''K''' is defined~~ to be
	~~:<math>\|x\|_{0} := \begin{cases} 0, & \text{if } x = 0 \\ 1, & \text{if } x \ne 0. \end{cases} </math>~~

	~~The '''real absolute value''' on the [[rational numbers\|rationals]] '''Q''' is the normal absolute value on the [[real numbers\|reals]], defined to be~~
	~~:<math>\|x\|_\infty := \begin{cases} x, & \text{if } x \ge 0 \\ -x, & \text{if } x < 0. \end{cases} </math>~~
	~~This~~ is ~~sometimes written with~~ a ~~subscript 1 instead of infinity~~.

	~~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~~
	~~:<math>\|x\|_{p} := \begin{cases} 0, & \text{if } x = 0 \\ p^{-n}, & \text{if } x \ne 0. \end{cases} </math>~~

	~~== Proof ==~~
	~~{{unreferenced section\|date=June 2013}}~~
	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
	~~<math>\|m/n\|_\ast~~
	~~=\|m\|_\ast/\|n\|_\ast~~
	~~=\|m\|^c_{\ast\ast}/\|n\|^c_{\ast\ast}~~
	~~=(\|m\|_{\ast\ast}/\|n\|_{\ast\ast})^c~~
	~~=\|m/n\|^c_{\ast\ast}</math>;~~
	~~and for negative rationals <math>\|{~~-~~}x\|_\ast=\|x\|_\ast=\|x\|^c_\infty=\|{-}x\|^c_\infty</math>.~~

	~~=== Case I~~: ~~∃''n'' ∈ '''N'''   \|''n''\| > 1 ===~~
	~~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:

	:~~<math> \begin{align}~~
	~~\|b\|_\ast^n = \|b^{n}\|_{\ast} &\leq am\max\{\|a\|_\ast^m,1\}\\~~
	~~&\leq a(n\log_a b+1)\max\{\|a\|_\ast^{n\log_a b},1\}\\~~
	~~\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\}\\~~
	~~\Rightarrow \|b\|_{\ast} &\leq \max\{\|a\|_\ast^{\log_{a}b},1\}.~~
	~~\end{align}</math>~~

	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.

	~~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>,~~
	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.

	~~=== Case II: ∀''n'' ∈ '''N'''   \|''n''\| ≤ 1 ===~~
	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.

	~~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.

	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.
	~~{{NumBlk\|1=\|2=\|3=<math>\blacksquare</math>\|RawN=~~.}}

	~~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>.~~

	~~== Another Ostrowski's theorem ==~~
	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>

	~~== See also ==~~
	* [[Valuation (algebra)]]
	* [[Absolute value]] in general

	~~== References ==~~
	~~{{reflist}}~~
	* {{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 }}~~
	*{{cite book \| last=Janusz \| first=Gerald J.
	~~\| title = Algebraic Number Fields~~
	~~\| edition = 2nd~~
	~~\| publisher = American Mathematical Society~~
	~~\| year = 1996, 1997~~
	~~\| isbn = 0-8218-0429-4}}~~
	*{{cite book \| last=Jacobson \| first=Nathan \| authorlink = Nathan Jacobson
	~~\| title = Basic algebra II~~
	~~\| edition = 2nd~~
	~~\| year = 1989~~
	~~\| publisher = W H Freeman~~
	~~\| isbn = 0-7167-1933-9}}~~
	*{{cite journal \| last=Ostrowski \| first=Alexander \| authorlink = Alexander Ostrowski
	~~\| title = Über einige Lösungen der Funktionalgleichung φ(x)·φ(y)=φ(xy)~~
	~~\| edition = 2nd~~
	~~\| year = 1916~~
	~~\| journal = Acta Mathematica~~
	~~\| issn = 0001-5962~~
	~~\| volume = 41~~
	~~\| issue = 1~~
	~~\| pages = 271–284~~
	~~\| url =~~ http://~~www~~.~~springerlink~~.~~com~~/~~content~~/~~96042g7576003r71/~~
	~~\| doi = 10.1007/BF02422947}}~~

	~~[[Category:Theorems in number theory]~~]

en>Vadmium: /* References */ Category:Duality theories sorting

2011-09-11T13:59:38Z

References: Category:Duality theories sorting

New page

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 | |p for some prime p or for p = ∞.}}</ref>

== Definitions ==
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
:<math>|x|_{\ast} = |x|^{c} \text{ for all } x \in \mathbf{K}.</math>

The '''trivial absolute value''' on any field '''K''' is defined to be
:<math>|x|_{0} := \begin{cases} 0, & \text{if } x = 0 \\ 1, & \text{if } x \ne 0. \end{cases} </math>

The '''real absolute value''' on the [[rational numbers|rationals]] '''Q''' is the normal absolute value on the [[real numbers|reals]], defined to be
:<math>|x|_\infty := \begin{cases} x, & \text{if } x \ge 0 \\ -x, & \text{if } x < 0. \end{cases} </math>
This is sometimes written with a subscript 1 instead of infinity.

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
:<math>|x|_{p} := \begin{cases} 0, & \text{if } x = 0 \\ p^{-n}, & \text{if } x \ne 0. \end{cases} </math>

== Proof ==
{{unreferenced section|date=June 2013}}
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
<math>|m/n|_\ast
=|m|_\ast/|n|_\ast
=|m|^c_{\ast\ast}/|n|^c_{\ast\ast}
=(|m|_{\ast\ast}/|n|_{\ast\ast})^c
=|m/n|^c_{\ast\ast}</math>;
and for negative rationals <math>|{-}x|_\ast=|x|_\ast=|x|^c_\infty=|{-}x|^c_\infty</math>.

=== Case I: ∃''n'' ∈ '''N'''   |''n''| > 1 ===
Consider the following calculation. Let <math>a,b\in\mathbf{N},>1</math>. Let <math>n\in\mathbf{N},>0</math>. Expressing ''b''''n'' 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:

:<math> \begin{align}
|b|_\ast^n = |b^{n}|_{\ast} &\leq am\max\{|a|_\ast^m,1\}\\
&\leq a(n\log_a b+1)\max\{|a|_\ast^{n\log_a b},1\}\\
\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\}\\
\Rightarrow |b|_{\ast} &\leq \max\{|a|_\ast^{\log_{a}b},1\}.
\end{align}</math>

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.

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>,
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.

=== Case II: ∀''n'' ∈ '''N'''   |''n''| ≤ 1 ===
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''''j''. We claim than in fact, that this is so for ''only'' one.

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.

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.
{{NumBlk|1=|2=|3=<math>\blacksquare</math>|RawN=.}}

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>.

== Another Ostrowski's theorem ==
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>

== See also ==
* [[Valuation (algebra)]]
* [[Absolute value]] in general

== References ==
{{reflist}}
* {{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 }}
*{{cite book | last=Janusz | first=Gerald J.
| title = Algebraic Number Fields
| edition = 2nd
| publisher = American Mathematical Society
| year = 1996, 1997
| isbn = 0-8218-0429-4}}
*{{cite book | last=Jacobson | first=Nathan | authorlink = Nathan Jacobson
| title = Basic algebra II
| edition = 2nd
| year = 1989
| publisher = W H Freeman
| isbn = 0-7167-1933-9}}
*{{cite journal | last=Ostrowski | first=Alexander | authorlink = Alexander Ostrowski
| title = Über einige Lösungen der Funktionalgleichung φ(x)·φ(y)=φ(xy)
| edition = 2nd
| year = 1916
| journal = Acta Mathematica
| issn = 0001-5962
| volume = 41
| issue = 1
| pages = 271–284
| url = http://www.springerlink.com/content/96042g7576003r71/
| doi = 10.1007/BF02422947}}

[[Category:Theorems in number theory]]

← Older revision		Revision as of 12:02, 5 May 2014
Line 1:		Line 1:
	~~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>~~		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.<br><br>Feel free to visit my web-site :: [http://srgame.co.kr/qna/21862 http://srgame.co.kr/qna/21862]

	~~== Definitions ==~~
	~~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~~
	~~:<math>\|x\|_{\ast} = \|x\|^{c} \text{ for all } x \in \mathbf{K}~~.~~</math>~~

	~~The '''trivial absolute value''' on any field '''K''' is defined~~ to be
	~~:<math>\|x\|_{0} := \begin{cases} 0, & \text{if } x = 0 \\ 1, & \text{if } x \ne 0. \end{cases} </math>~~

	~~The '''real absolute value''' on the [[rational numbers\|rationals]] '''Q''' is the normal absolute value on the [[real numbers\|reals]], defined to be~~
	~~:<math>\|x\|_\infty := \begin{cases} x, & \text{if } x \ge 0 \\ -x, & \text{if } x < 0. \end{cases} </math>~~
	~~This~~ is ~~sometimes written with~~ a ~~subscript 1 instead of infinity~~.

	~~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~~
	~~:<math>\|x\|_{p} := \begin{cases} 0, & \text{if } x = 0 \\ p^{-n}, & \text{if } x \ne 0. \end{cases} </math>~~

	~~== Proof ==~~
	~~{{unreferenced section\|date=June 2013}}~~
	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
	~~<math>\|m/n\|_\ast~~
	~~=\|m\|_\ast/\|n\|_\ast~~
	~~=\|m\|^c_{\ast\ast}/\|n\|^c_{\ast\ast}~~
	~~=(\|m\|_{\ast\ast}/\|n\|_{\ast\ast})^c~~
	~~=\|m/n\|^c_{\ast\ast}</math>;~~
	~~and for negative rationals <math>\|{~~-~~}x\|_\ast=\|x\|_\ast=\|x\|^c_\infty=\|{-}x\|^c_\infty</math>.~~

	~~=== Case I~~: ~~∃''n'' ∈ '''N'''   \|''n''\| > 1 ===~~
	~~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:

	:~~<math> \begin{align}~~
	~~\|b\|_\ast^n = \|b^{n}\|_{\ast} &\leq am\max\{\|a\|_\ast^m,1\}\\~~
	~~&\leq a(n\log_a b+1)\max\{\|a\|_\ast^{n\log_a b},1\}\\~~
	~~\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\}\\~~
	~~\Rightarrow \|b\|_{\ast} &\leq \max\{\|a\|_\ast^{\log_{a}b},1\}.~~
	~~\end{align}</math>~~

	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.

	~~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>,~~
	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.

	~~=== Case II: ∀''n'' ∈ '''N'''   \|''n''\| ≤ 1 ===~~
	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.

	~~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.

	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.
	~~{{NumBlk\|1=\|2=\|3=<math>\blacksquare</math>\|RawN=~~.}}

	~~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>.~~

	~~== Another Ostrowski's theorem ==~~
	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>

	~~== See also ==~~
	* [[Valuation (algebra)]]
	* [[Absolute value]] in general

	~~== References ==~~
	~~{{reflist}}~~
	* {{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 }}~~
	*{{cite book \| last=Janusz \| first=Gerald J.
	~~\| title = Algebraic Number Fields~~
	~~\| edition = 2nd~~
	~~\| publisher = American Mathematical Society~~
	~~\| year = 1996, 1997~~
	~~\| isbn = 0-8218-0429-4}}~~
	*{{cite book \| last=Jacobson \| first=Nathan \| authorlink = Nathan Jacobson
	~~\| title = Basic algebra II~~
	~~\| edition = 2nd~~
	~~\| year = 1989~~
	~~\| publisher = W H Freeman~~
	~~\| isbn = 0-7167-1933-9}}~~
	*{{cite journal \| last=Ostrowski \| first=Alexander \| authorlink = Alexander Ostrowski
	~~\| title = Über einige Lösungen der Funktionalgleichung φ(x)·φ(y)=φ(xy)~~
	~~\| edition = 2nd~~
	~~\| year = 1916~~
	~~\| journal = Acta Mathematica~~
	~~\| issn = 0001-5962~~
	~~\| volume = 41~~
	~~\| issue = 1~~
	~~\| pages = 271–284~~
	~~\| url =~~ http://~~www~~.~~springerlink~~.~~com~~/~~content~~/~~96042g7576003r71/~~
	~~\| doi = 10.1007/BF02422947}}~~

	~~[[Category:Theorems in number theory]~~]