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 then:
 * $\norm {m a + n b} = 1$

By Corollary 5 of Characterisation of Non-Archimedean Division Ring Norms then:
 * $\norm a, \norm b, \norm n, \norm m \le 1$

Let $\norm a \lt 1$.

By Norm axiom $(\text N 2)$: Multiplicativity:
 * $\norm {m a} = \norm m \norm a \lt 1$

Hence:
 * $\norm {m a} < \norm {m a + n b}$

By 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.