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.
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
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
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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 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|∗ > 1
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:
However we have:
Now choose 1 < b ∈ N 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
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|∗ ≤ 1
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, n ∈ Z such that . This yields
So we must have |pj|∗ = α < 1 for some j, and |pi|∗ = 1 for i ≠ j. 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 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.
- 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 = ∞.
- Cassels (1986) p. 33
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- Ostrowski, Alexander (1916). "Über einige Lösungen der Funktionalgleichung φ(x)·φ(y)=φ(xy)". Acta Mathematica (2nd ed.) 41 (1): 271–284. doi:10.1007/BF02422947. ISSN 0001-5962.