Three Points in Ultrametric Space have Two Equal Distances/Corollary 5

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 $1.50$