Characterisation of Non-Archimedean Division Ring Norms/Corollary 1

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 $2.2.3$