Characterisation of Non-Archimedean Division Ring Norms/Necessary Condition

Theorem

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with unity $1_R$.

Then:

$\norm {\,\cdot\,}$ is non-Archimedean $\implies \forall n \in \N_{>0}: \norm {n \cdot 1_R} \le 1$.

where: $n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} }$

Let $\norm {\,\cdot\,}$ be non-Archimedean.

Then by the definition of a non-Archimedean norm, for $n \in \N$:

$\ds \forall n \in \N_{>0}: \, $

$\ds \norm {n \cdot 1_R}$

$=$

$\ds \norm {1_R + \dots + 1_R}$

($n$ summands)

$\ds $

$\le$

$\ds \max \set {\norm {1_R}, \ldots, \norm {1_R} }$

$\ds $

$=$

$\ds 1$

because $\norm {1_R} = 1$

$\blacksquare$

2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.2$ Normed Fields, Proposition $1.14$
1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 2.2$ Basic Properties, Theorem $2.2.2$