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


$\blacksquare$


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