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In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.[1]

Contents

DefinitionsEdit

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 Q 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 Q is defined as follows: any non-zero rational x, can be written uniquely as  , where a and b are integers not divisible by p and n is an integer; so we define

 

ProofEdit

Consider a non-trivial absolute value on the rationals  . We consider two cases,

(i)  
(ii)  

It suffices for us to consider the valuation of integers greater than one. For, if we find c in R+ 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

 

Case I: ∃n ∈ N   |n| > 1Edit

Consider the following calculation. Let a, b and n be natural numbers with a, b > 1. Expressing bn in base a yields

 

Then we see, by the properties of an absolute value:

 

Therefore,

 

However we have:

 

which implies:

 

Now choose 1 < bN such that |b| > 1. Using this in the above ensures that |a| > 1 regardless of the choice of a (else   implying  ). Thus for any choice of a, b > 1 above, we get

 

i.e.

 

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.

Case II: ∀n ∈ N   |n| ≤ 1Edit

As this valuation is non-trivial, there must be a natural number for which |n| < 1. Factorising this natural,

 

yields |p| must be less than 1, for at least one of the prime factors p = pj. We claim than in fact, that 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  . By the Euclidean algorithm, there are m, nZ such that  . This yields

 

a contradiction.

So we must have |pj| = α < 1 for some j, and |pi| = 1 for ij. Letting

 

we see that for general positive naturals

 

As per the above remarks we see that   all rationals, implying 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 theoremEdit

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.[2]

See alsoEdit

ReferencesEdit

  1. ^ Koblitz, Neal (1984). P-adic numbers, p-adic analysis, and zeta-functions (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 ℚ is equivalent to | |p for some prime p or for p = ∞.
  2. ^ Cassels (1986) p. 33