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:
- $\sup \set {\norm {n \cdot 1_R}: n \in \N_{> 0}} = 1$.
where $n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$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.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.2$ Basic Properties: Corollary $2.2.3$