Ostrowski's theorem

Ostrowski's theorem, due to Alexander Ostrowski, states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.

Two absolute values | | and | |^* on a field F are defined to be equivalent if there exists a real number $i>0$ such that

$|x|^{*}=|x|^{i}{\mbox{ for all }}x\in F.$

The trivial absolute value on any field F is defined to be

$|x|_{0}:={\begin{cases}0,&{\mbox{if }}x=0\\1,&{\mbox{if }}x\neq 0.\end{cases}}$

The real absolute value on Q is the normal absolute value on the real numbers, defined to be

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

For a prime number p, the p-adic absolute value on Q is defined as follows: any non-zero rational number x, can be written uniquely as $x=p^{n}{\frac {a}{b}}$ with a, b and p pairwise coprime and where n can be positive, negative or 0; then

$|x|_{p}:={\begin{cases}0,&{\mbox{if }}x=0\\p^{-n},&{\mbox{if }}x\neq 0.\end{cases}}$

Other theorems referred to as Ostrowski's theorem

Another theorem states that any field complete with respect to an Archimedean absolute value is (algebraically and topologically) isomorphic to either the real numbers or the complex numbers. This is often called Ostrowski's theorem.

See also

• Valuation (mathematics)
• Absolute value

References

• Gerald J. Janusz (1996, 1997). Algebraic Number Fields (2^nd edition ed.). American Mathematical Society. ISBN 0-8218-0429-4. {{cite book}}: |edition= has extra text (help); Check date values in: |year= (help)
• Nathan Jacobson (1989). Basic algebra II (2^nd ed. ed.). W H Freeman. ISBN 0-7167-1933-9. {{cite book}}: |edition= has extra text (help)