On all absolute values of rational numbers
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.
Raising an absolute value to a power less than 1 always results in another absolute value. Two absolute values and on a field K are defined to be equivalent if there exists a real number c > 0 such that
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
Consider a non-trivial absolute value on the rationals . We consider two cases:
It suffices for us to consider the valuation of integers greater than one. For, if we find for which for all naturals greater than one, then this relation trivially holds for 0 and 1, and for positive rationals
and for negative rationals
This case implies that there exists such that By the properties of an absolute value, and , so (it cannot be zero). It therefore follows that b > 1.
Now, let with a > 1. Express bn in base a:
Then we see, by the properties of an absolute value:
As each of the terms in the sum is smaller than , it follows:
However, as , we have
Together with the condition the above argument shows that regardless of the choice of a > 1 (otherwise , implying ). As a result, the initial condition above must be satisfied by any b > 1.
Thus for any choice of natural numbers a, b > 1, we get
By symmetry, this inequality is an equality.
Since a, b were arbitrary, there is a constant for which , i.e. for all naturals n > 1. As per the above remarks, we easily see that for all rationals, thus demonstrating equivalence to the real absolute value.
As this valuation is non-trivial, there must be a natural number for which Factoring into primes:
yields that there exists such that We claim that in fact this is so for only one.
Suppose per contra that p, q are distinct primes with absolute value less than 1. First, let be such that . Since and are coprime, there are such that This yields
So we must have for some j, and for i ≠ j. Letting
we see that for general positive naturals
As per the above remarks, we see that for all rationals, implying that the absolute value is equivalent to the p-adic one.
One can also show a stronger conclusion, namely, that is a nontrivial absolute value if and only if either for some or for some .
Another 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 sometimes also referred to as Ostrowski's theorem.