Characterisation of Non-Archimedean Division Ring Norms/Necessary Condition

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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 $n \in \N$,

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


$\blacksquare$

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