Ostrowski's theorem

The set of Ostrowski is a theorem from the mathematical field of number theory ( reviewed theory ). It says that every nontrivial absolute amount defined on the rational numbers is equivalent either to the usual amount function or to a p-adic amount . The theorem was proven by Alexander Ostrowski in 1916 .

Definitions

An absolute amount on a body is a picture ${\displaystyle K}$

${\displaystyle \vert .\vert \colon K\to \mathbb {R} }$

${\displaystyle x\mapsto \vert x\vert }$,

which fulfills the following properties for all : ${\displaystyle x,y\in K}$

${\displaystyle \vert x\vert \geq 0}$,

${\displaystyle \vert x\vert =0\;\Leftrightarrow \;x=0}$,

${\displaystyle \vert xy\vert =\vert x\vert \vert y\vert }$,

${\displaystyle \vert x+y\vert \leq \vert x\vert +\vert y\vert }$.

Two absolute values and are called equivalent if there is a real number with ${\displaystyle \vert .\vert }$ ${\displaystyle \vert .\vert _{*}}$ ${\displaystyle c>0}$

${\displaystyle \vert x\vert ^{c}=\vert x\vert _{*}}$

for everyone . ${\displaystyle x\in K}$

Examples

The trivial amount :

${\displaystyle |x|_{0}:={\begin{cases}0,&{\text{für }}x=0\\1,&{\text{für }}x\neq 0.\end{cases}}}$.

The amount function defined on the real numbers

${\displaystyle |x|_{\infty }:={\begin{cases}x,&{\text{für }}x\geq 0\\-x,&{\text{für }}x<0.\end{cases}}}$.

The p-adic amount defined for each prime number on the rational numbers : ${\displaystyle p}$

${\displaystyle |x|_{p}:={\begin{cases}0,&{\text{für }}x=0\\p^{-n},&{\text{für }}x=p^{n}{\frac {a}{b}}{\text{ mit }}a,b,p{\text{ paarweise teilerfremd und }}n\in \mathbb {Z} .\end{cases}}}$

Here it is used that every rational number can be represented uniquely as with pairwise prime numbers and . ${\displaystyle x=p^{n}{\frac {a}{b}}}$ ${\displaystyle p,a,b}$ ${\displaystyle n\in \mathbb {Z} }$

Ostrowski's theorem

Any nontrivial absolute amount defined on the rational numbers ${\displaystyle \mathbb {Q} }$

${\displaystyle \vert .\vert \colon \mathbb {Q} \to \mathbb {R} }$

is equivalent either to the amount function or to a p-adic amount . ${\displaystyle \vert .\vert _{\infty }}$ ${\displaystyle \vert .\vert _{p}}$

literature

• Alexander Ostrowski: About some solutions of the functional equation ψ (x) ⋅ψ (x) = ψ (xy). Acta Math. 41 (1916), no. 1, 271-284.
• Emil Artin : Algebraic numbers and algebraic functions. I. Institute for Mathematics and Mechanics, New York University, New York, 1951.
• Edwin Weiss: Algebraic number theory. McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London 1963.
• ACM van Rooij: Non-Archimedean functional analysis. Monographs and Textbooks in Pure and Applied Math., 51. Marcel Dekker, Inc., New York, 1978. ISBN 0-8247-6556-7 .
• Fernando Q. Gouvêa: p-adic numbers. An introduction. University text. Springer-Verlag, Berlin, 1993. ISBN 3-540-56844-1 .