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 $n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{n \, times}$

Proof

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

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

 $\, \displaystyle \forall n \in \N_{>0}: \,$ $\displaystyle \norm{n \cdot 1_R}$ $=$ $\displaystyle \norm{1_R + \dots + 1_R}$ ($n$ summands) $\displaystyle$ $\le$ $\displaystyle \max \set{ \norm{1_R}, \ldots, \norm{1_R} }$ $\displaystyle$ $=$ $\displaystyle 1$ because $\norm{1_R} = 1$

$\blacksquare$