Ostrowski's Theorem/Non-Archimedean Norm
Theorem
Let $\norm {\, \cdot \,}$ be a non-trivial non-Archimedean norm on the rational numbers $\Q$.
Then $\norm {\, \cdot \,}$ is equivalent to the $p$-adic norm $\norm {\, \cdot \,}_p$ for some prime $p$.
Proof
From Characterisation of Non-Archimedean Division Ring Norms then:
- $\forall n \in \N: \norm n \le 1$
Lemma 2.1
- $\exists n \in \N: 0 < \norm n < 1$.
$\Box$
Let $n_0 = \min \set {n \in N : \norm n < 1}$.
Lemma 2.2
- $n_0$ is a prime number.
$\Box$
Let $p = n_0$.
Let $\alpha = - \dfrac {\log \norm p } {\log p}$ then:
- $\norm p = p^{-\alpha} = \paren {p^{-1}}^\alpha = \norm p_p^\alpha$
Let $b \in N$
Case 1: $p \nmid b$
Let $p \nmid b$.
From Prime not Divisor implies Coprime:
- $p$ and $b$ are coprime, that is, $p \perp b$
From Corollary 5 of Three Points in Ultrametric Space have Two Equal Distances:
- $\norm b = 1$
By the definition of the $p$-adic norm:
- $\norm b_p = 1$
Hence:
- $\norm b = 1 = 1^\alpha = \norm b_p^\alpha$
$\Box$
Case 2: $p \divides b$
Let $p \divides b$.
Let $\nu = \map {\nu_p} b$ where $\nu_p$ is the $p$-adic valuation on $\Z$.
Then:
- $b = p^\nu c$
where $p \nmid c$
From #Case 1:
- $\norm c = 1$
Hence:
\(\ds \norm b\) | \(=\) | \(\ds \norm p^\nu \norm {c}\) | Non-Archimedean Norm Axiom $\text N 2$: Multiplicativity | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm p^\nu\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \norm p_p^{\alpha \nu}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {p^{-1} }^{\alpha \nu}\) | Definition of $p$-adic norm | |||||||||||
\(\ds \) | \(=\) | \(\ds p^{-\alpha \nu}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {p^{-\nu} }^\alpha\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \norm b_p^\alpha\) | Definition of $p$-adic norm |
$\Box$
In either case:
- $\norm b = \norm b_p^\alpha$
Since $b$ was arbitrary, it has been shown:
- $\forall b \in \N: \norm b = \norm b_p^\alpha$
From Equivalent Norms on Rational Numbers:
- $\norm {\, \cdot \,}$ is equivalent to the $p$-adic norm $\norm {\, \cdot \,}_p$.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.1$ Absolute Values on $\Q$, Theorem $3.1.3$
- 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.9$ Metrics and norms on the rational numbers. Ostrowski’s Theorem, Theorem $1.50$