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.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 2.2$ Basic Properties, Theorem $2.2.2$
- 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.2$ Normed Fields, Proposition $1.14$