Ostrowski's theorem

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 {displaystyle mathbb {Q} } is equivalent to either the usual real absolute value or a p-adic absolute value.[1] Contents 1 Definitions 2 Proof 2.1 Case (1) 2.2 Case (2) 3 Another Ostrowski's theorem 4 See also 5 References Definitions Raising an absolute value to a power less than 1 always results in another absolute value. Two absolute values {displaystyle |cdot |} and {displaystyle |cdot |_{*}} on a field K are defined to be equivalent if there exists a real number c > 0 such that {displaystyle forall xin K:quad |x|_{*}=|x|^{c}.} The trivial absolute value on any field K is defined to be {displaystyle |x|_{0}:={begin{cases}0&x=0,\1&xneq 0.end{cases}}} The real absolute value on the rationals {displaystyle mathbb {Q} } is the standard absolute value on the reals, defined to be {displaystyle |x|_{infty }:={begin{cases}x&xgeq 0,\-x&x<0.end{cases}}} This is sometimes written with a subscript 1 instead of infinity. For a prime number p, the p-adic absolute value on {displaystyle mathbb {Q} } is defined as follows: any non-zero rational x can be written uniquely as {displaystyle x=p^{n}{tfrac {a}{b}}} , where a and b are coprime integers not divisible by p, and n is an integer; so we define {displaystyle |x|_{p}:={begin{cases}0&x=0,\p^{-n}&xneq 0.end{cases}}} Proof This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (June 2013) (Learn how and when to remove this template message) Consider a non-trivial absolute value on the rationals {displaystyle (mathbb {Q} ,|cdot |_{*})} . We consider two cases: {displaystyle {begin{aligned}(1)quad exists nin mathbb {N} qquad |n|_{*}&>1,\(2)quad forall nin mathbb {N} qquad |n|_{*}&leq 1.end{aligned}}} It suffices for us to consider the valuation of integers greater than one. For, if we find {displaystyle cin mathbb {R} _{+}} for which {displaystyle |n|_{*}=|n|_{**}^{c}} for all naturals greater than one, then this relation trivially holds for 0 and 1, and for positive rationals {displaystyle left|{frac {m}{n}}right|_{*}={frac {|m|_{*}}{|n|_{*}}}={frac {|m|_{**}^{c}}{|n|_{**}^{c}}}=left({frac {|m|_{**}}{|n|_{**}}}right)^{c}=left|{frac {m}{n}}right|_{**}^{c},} and for negative rationals {displaystyle |-x|_{*}=|x|_{*}=|x|_{**}^{c}=|-x|_{**}^{c}.} Case (1) This case implies that there exists {displaystyle bin mathbb {N} } such that {displaystyle |b|_{*}>1.} By the properties of an absolute value, {displaystyle |0|_{*}=0} and {displaystyle |1|_{*}^{2}=|1|_{*}} , so {displaystyle |1|_{*}=1} (it cannot be zero). It therefore follows that b > 1.

Now, let {displaystyle a,nin mathbb {N} } with a > 1. Express bn in base a: {displaystyle b^{n}=sum _{i0.} Hence {displaystyle b^{n}geq a^{m-1},quad } so {displaystyle quad mleq n,{frac {log b}{log a}}+1.} Then we see, by the properties of an absolute value: {displaystyle |b|_{*}^{n}=|b^{n}|_{*}leq sum _{i1,} the above argument shows that {displaystyle |a|_{*}>1} regardless of the choice of a > 1 (otherwise {displaystyle |a|_{*}^{log _{a}!b}leq 1} , implying {displaystyle |b|_{*}leq 1} ). As a result, the initial condition above must be satisfied by any b > 1.

Thus for any choice of natural numbers a, b > 1, we get {displaystyle |b|_{*}leq |a|_{*}^{frac {log b}{log a}},} i.e.

{displaystyle {frac {log |b|_{*}}{log b}}leq {frac {log |a|_{*}}{log a}}.} By symmetry, this inequality is an equality.

Since a, b were arbitrary, there is a constant {displaystyle lambda in mathbb {R} _{+}} for which {displaystyle log |n|_{*}=lambda log n} , i.e. {displaystyle |n|_{*}=n^{lambda }=|n|_{infty }^{lambda }} for all naturals n > 1. As per the above remarks, we easily see that {displaystyle |x|_{*}=|x|_{infty }^{lambda }} 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 {displaystyle |n|_{*}<1.} Factoring into primes: {displaystyle n=prod _{i

Si quieres conocer otros artículos parecidos a Ostrowski's theorem puedes visitar la categoría Theorems in algebraic number theory.

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