Ostrowski's theorem
In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers formula_1 is equivalent to either the usual real absolute value or a -adic absolute value. Definitions. Raising an absolute value to a power less than 1 always results in another absolute value. Two absolute values formula_2 and formula_3 on a field "K" are defined to be equivalent if there exists a real number such that The trivial absolute value on any field "K" is defined to be The real absolute value on the rationals formula_1  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 formula_1 is defined as follows: any non-zero rational can be written uniquely as formula_9, where and are coprime integers not divisible by , and is an integer; so we define Proof. Consider a non-trivial absolute value on the rationals formula_11. We consider two cases: It suffices for us to consider the valuation of integers greater than one. For, if we find formula_13 for which formula_14 for all naturals  greater than one, then this relation trivially holds for 0 and 1, and for positive rationals and for negative rationals Case (1). This case implies that there exists formula_17 such that formula_18 By the properties of an absolute value, formula_19 and formula_20, so formula_21 (it cannot be zero). It therefore follows that . Now, let formula_22 with . Express in base : Hence Then we see, by the properties of an absolute value: As each of the terms in the sum is smaller than formula_27, it follows: Therefore, However, as formula_30, we have which implies Together with the condition formula_33 the above  argument shows that formula_34 regardless of the choice of (otherwise formula_35, implying formula_36). As a result, the initial condition above must be satisfied by any . Thus for any choice of natural numbers , we get i.e. By symmetry, this inequality is an equality. Since were arbitrary, there is a constant formula_39 for which formula_40, i.e. formula_41 for all naturals . As per the above remarks, we easily see that formula_42 for all rationals, thus demonstrating equivalence to the real absolute value. Case (2). As this valuation is non-trivial, there must be a natural number for which formula_43 Factoring into primes: yields  that there exists formula_45 such that formula_46 We claim that in fact this is so for "only" one. Suppose "per contra" that are distinct primes with absolute value less than 1. First, let formula_47 be such that formula_48. Since formula_49 and formula_50 are coprime, there are formula_51 such that formula_52 This yields a contradiction. So we must have formula_54 for some , and formula_55 for . Letting we see that for general positive naturals As per the above remarks, we see that formula_58 for all rationals, implying that the absolute value is equivalent to the -adic one. formula_59 One can also show  a stronger conclusion, namely, that formula_60 is a nontrivial absolute value if and only if either formula_61 for some formula_62 or formula_63 for some formula_64. 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.