Characterisation of Non-Archimedean Division Ring Norms/Corollary 1

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Theorem

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


$\norm{\,\cdot\,}$ is non-Archimedean if and only if:

$\sup \set {\norm{n \cdot 1_R}: n \in \N_{\gt 0}} = 1$.


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

Proof

By Characterisation of Non-Archimedean Division Ring Norms then:

$\norm{\,\cdot\,}$ is non-Archimedean if and only if $\,\,\sup \set {\norm{n \cdot 1_R}: n \in \N_{\gt 0}} \le 1$.

By norm of unity then:

$\norm {1_R} = 1$

The result follows.

$\blacksquare$

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