Characterisation of Non-Archimedean Division Ring Norms

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

Then $\norm{\,\cdot\,}$ is non-archimedean :


 * $\forall n \in \N_{>0}: \norm{n \cdot 1_R} \le 1$.

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

Necessary Condition
Let $\norm{\,\cdot\,}$ is non-archimedean.

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

Sufficient Condition
Let:
 * $\forall n \in \N_{>0}: \norm{n \cdot 1_R} \le 1$

We compute:

Taking $n$th roots yields:
 * $\left\vert{x + y}\right\vert \le \max \left\{ {\left\vert{x}\right\vert, \left\vert{y}\right\vert}\right\}$