Ostrowski's theorem

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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

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

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

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 with a, b and p pairwise coprime and where n can be positive, negative or 0; then

[edit] Other theorems referred to as Ostrowski's theorem

Since the only field extensions of the real numbers are the real numbers and the complex numbers, we can deduce as a corollary of Ostrowski's theorem that any field complete with respect to an Archimedean absolute value is isomorphic to either the real numbers or the complex numbers. This is often called Ostrowski's theorem, although it is weaker than the above theorem as it considers only the case that the absolute value is Archimedean.

[edit] See also

• Absolute value (algebra)
• Valuation (mathematics)
• Absolute value

[edit] References

• Gerald J. Janusz (1996, 1997). Algebraic Number Fields, 2^nd edition, American Mathematical Society. ISBN 0-8218-0429-4. 
• Nathan Jacobson (1989). Basic algebra II, 2^nd ed., W H Freeman. ISBN 0-7167-1933-9.