Characterisation of Non-Archimedean Division Ring Norms/Corollary 3
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Theorem
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with unity $1_R$.
$\norm {\,\cdot\,}$ is Archimedean if and only if:
- $\sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } = +\infty$
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:
- $\norm {\,\cdot\,}$ is Archimedean $\iff \sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } > 1$
By Corollary 2:
- $\sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } > 1 \iff \sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } = +\infty$
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.2$ Basic Properties: Theorem $2.2.2$
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$ Normed Fields: Proposition $1.14$