Ostrowski's theorem
In number theory, Ostrowski's theorem, due to Alexander Ostrowski, states that every non-trivial absolute value on the rational numbers is equivalent to either the usual real absolute value or a -adic absolute value.
Definitions
An absolute value on the rational numbers is a function satisfying for all that,,, and only if.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
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, the -adic absolute value on is defined as follows: any non-zero rational can be written uniquely as, where and are coprime integers not divisible by, and is an integer; so we define
Theorem statement
Let be any absolute value on the rational numbers. Then either, or is equivalent to, or is equivalent to.Proof
The following proof follows the one of Theorem 10.1 in Schikhof.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, the multiplicativity property entails. In other words, the absolute value of a negative integer coincides with that of its opposite.
Let be a positive integer. From the fact that and the multiplicativity property, we conclude that.
Let now be a positive rational. There exist two coprime positive integers and 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, we can write
where is the p-adic valuation of. The multiplicativity property enables one to compute the absolute value of from that of the prime numbers using the following relationship
We continue the proof by separating two cases:
- There exists a positive integer such that ; or
- For all integer, one has.
First case
Suppose that there exists a positive integer such that Let be a non-negative integer and be a positive integer greater than. We express in base : there exist a positive integer and integers such that for all, and. In particular, so.Each term is at most. Besides, is at most. By the triangle inequality and the above bound on, it follows:
Therefore, raising both sides to the power, we obtain
Finally, taking the limit as tends to infinity shows that
Together with the condition the above argument leads to regardless of the choice of . As a result, all integers greater than one have an absolute value strictly greater than one. Thus generalizing the above, for any choice of integers and greater than or equal to 2, we get
i.e.
By symmetry, this inequality is an equality. In particular, for all,, i.e.. Because the triangle inequality implies that for all positive integers we have, in this case we obtain more precisely that.
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, thus demonstrating equivalence to the real absolute value.
Second case
Suppose that for all integer, one has. As our absolute value is non-trivial, there must exist a positive integer for which Decomposing on the prime numbers shows that there exists such that. We claim that in fact this is so for one prime number only.Suppose by way of contradiction that and are two distinct primes with absolute value strictly less than 1. Let be a positive integer such that and are smaller than. By Bézout's identity, since and are coprime, there exist two integers and such that This yields a contradiction, as
This means that there exists a unique prime such that and that for all other primes, one has . Let. From, we infer that.
We finally verify that and that for all other primes,. As per the above result on the determination of an absolute value by its values on the prime numbers, we conclude that for all rational, implying that this absolute value is equivalent to the -adic one. [Q.E.D.|]