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.[1]

Definitions

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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.[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

 

Proof

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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. 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:

  1. There exists a positive integer n such that  ; or
  2. For all integer n, one has  .

First case

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Suppose that there exists a positive integer n such that   Let k be a non-negative integer and b be a positive integer greater than  . We express   in base b: there exist a positive integer m and integers   such that for all i,   and  . In particular,   so  .

Each term   is smaller than  . (By the multiplicative property,  , then using the fact that   is a digit, write   so by the triangle inequality,  .) Besides,   is smaller than  . 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   the above argument leads to   regardless of the choice of b (otherwise   implies  ). 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 n and b 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 n 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 r, thus demonstrating equivalence to the real absolute value.

Second case

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Suppose that for all integer n, one has  . As our absolute value is non-trivial, there must exist a positive integer n 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 per contra that p and q are two distinct primes with absolute value strictly less than 1. Let k be a positive integer such that   and   are smaller than  . By Bézout's identity, since   and   are coprime, there exist two integers a and b such that   This yields a contradiction, as

 

This means that there exists a unique prime p such that   and that for all other prime q, one has   (from the hypothesis of this second case). Let  . From  , we infer that  . (And indeed in this case, all positive   give absolute values equivalent to the p-adic one.)

We finally verify that   and that for all other prime q,  . 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 r, implying that this absolute value is equivalent to the p-adic one.  

Another Ostrowski's theorem

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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.[3]

See also

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References

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  1. ^ Koblitz, Neal (1984). P-adic numbers, p-adic analysis, and zeta-functions. Graduate Texts in Mathematics (2nd ed.). New York: Springer-Verlag. p. 3. ISBN 978-0-387-96017-3. Retrieved 24 August 2012. Theorem 1 (Ostrowski). Every nontrivial norm ‖ ‖ on   is equivalent to | |p for some prime p or for p = ∞.
  2. ^ Schikhof (2007) Theorem 9.2 and Exercise 9.B
  3. ^ Cassels (1986) p. 33