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Concept
Ostrowski's theorem
Summary
In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers is equivalent to either the usual real absolute value or a p-adic absolute value.
Two absolute values and on the rationals are defined to be equivalent if they induce the same topology; this can be shown to be equivalent to the existence of a positive real number such that
(Note: In general, if is an absolute value, is not necessarily an absolute value anymore; however if two absolute values are equivalent, then each is a positive power of the other.) The trivial absolute value on any field K is defined to be
The real absolute value on the rationals is the standard absolute value on the reals, defined to be
This is sometimes written with a subscript 1 instead of infinity.
For a prime number p, the p-adic absolute value on is defined as follows: any non-zero rational x can be written uniquely as , where a and b are coprime integers not divisible by p, and n is an integer; so we define
The following proof follows the one of Theorem 10.1 in Schikhof (2007).
Let be an absolute value on the rationals. We start the proof by showing that it is entirely determined by the values it takes on prime numbers.
From the fact that and the multiplicativity property of the absolute value, we infer that . In particular, has to be 0 or 1 and since , one must have . A similar argument shows that .
For all positive integer n, the multiplicativity property entails . In other words, the absolute value of a negative integer coincides with that of its opposite.
Let n be a positive integer. From the fact that and the multiplicativity property, we conclude that .
Let now r be a positive rational. There exist two coprime positive integers p and q such that . The properties above show that . Altogether, the absolute value of a positive rational is entirely determined from that of its numerator and denominator.
Finally, let be the set of prime numbers. For all positive integer n, we can write
where is the p-adic valuation of n.
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