# 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$times} }$

## Proof

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 $\norm {1_R} = 1$

$\blacksquare$