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.
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
is not necessarily an absolute value anymore; however if two absolute values are equivalent, then each is a positive power of the other.
[2]) 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).
We start the proof by showing that it is entirely determined by the values it takes on prime numbers.
and the multiplicativity property of the absolute value, we infer 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.
There exist two coprime positive integers p and q such that
Altogether, the absolute value of a positive rational is entirely determined from that of its numerator and denominator.
is the p-adic valuation of n. The multiplicativity property enables one to compute the absolute value of n from that of the prime numbers using the following relationship We continue the proof by separating two cases: Suppose that there exists a positive integer n such that
By the triangle inequality and the above bound on m, it follows: Therefore, raising both sides to the power
, we obtain Finally, taking the limit as k tends to infinity shows that Together with the condition
Thus generalizing the above, for any choice of integers n and b greater than or equal to 2, we get i.e. By symmetry, this inequality is an equality.
Because the triangle inequality implies that for all positive integers n we have
As per the above result on the determination of an absolute value by its values on the prime numbers, we easily see that
for all rational r, thus demonstrating equivalence to the real absolute value.
As our absolute value is non-trivial, there must exist a positive integer n for which
on the prime numbers shows that there exists
We claim that in fact this is so for one prime number only.
Suppose per contra that p and q are two distinct primes with absolute value strictly less than 1.
This yields a contradiction, as This means that there exists a unique prime p such that
give absolute values equivalent to the p-adic one.)
for all rational r, implying that this absolute value is equivalent to the p-adic one.
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.