Three Points in Ultrametric Space have Two Equal Distances/Corollary 5
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Theorem
Let $\norm {\, \cdot \,}$ be a non-trivial non-Archimedean norm on the rational numbers $\Q$.
Let $a, b \in \Z_{\ne 0}$ be coprime:
- $a \perp b$
Then:
- $\norm a = 1$ or $\norm b = 1$
Proof
By Bézout's Identity then:
- $\exists n, m \in \Z : m a + n b = 1$
By Norm of Unity:
- $\norm {m a + n b} = 1$
By Characterisation of Non-Archimedean Division Ring Norms: Corollary $5$:
- $\norm a, \norm b, \norm n, \norm m \le 1$
Let $\norm a < 1$.
By Norm Axiom $\text N 2$: Multiplicativity:
- $\norm {m a} = \norm m \norm a < 1$
Hence:
- $\norm {m a} < \norm {m a + n b}$
By Three Points in Ultrametric Space have Two Equal Distances: Corollary $4$:
- $\norm {n b} = \norm {m a + n b} = 1$
By Norm Axiom $\text N 2$: Multiplicativity:
- $\norm n \norm b = 1$
Hence:
- $\norm b = 1$
The result follows.
$\blacksquare$
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 $1.50$