Characterisation of Non-Archimedean Division Ring Norms/Corollary 5

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Theorem

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


If $\norm {\, \cdot \,}$ is non-Archimedean then:

$\sup \set {\norm {n \cdot 1_R}: n \in \Z} = 1$


where $n \cdot 1_R = \begin{cases} \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} } & : n > 0 \\ 0 & : n = 0 \\ \\ -\underbrace {\paren {1_R + 1_R + \dots + 1_R} }_{\text {$-n$ times} } & : n < 0 \\ \end{cases}$


Proof

By Corollary 1 of Characterisation of Non-Archimedean Division Ring Norms then:

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

By Norm Axiom $(\text N 1)$ (Positive Definiteness) then:

$\norm {0 \cdot 1_R} = 0 \le 1$


Let $n < 0$ then:

\(\ds \norm {n \cdot 1_R}\) \(=\) \(\ds \norm {-\underbrace {\paren {1_R + 1_R + \dots + 1_R} }_{\text {$-n$ times} } }\)
\(\ds \) \(=\) \(\ds \norm {\underbrace {1_R + 1_R + \dots + 1_R}_{\text {$-n$ times} } }\) Norm of Ring Negative
\(\ds \) \(=\) \(\ds \norm {\paren {-n} \cdot 1_R}\)
\(\ds \) \(\le\) \(\ds 1\) Characterisation of Non-Archimedean Division Ring Norms


The result follows.

$\blacksquare$


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