Characterisation of Non-Archimedean Division Ring Norms/Corollary 4

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Theorem

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

Let $R$ have characteristic $p>0$.


Then $\norm{\,\cdot\,}$ is a non_Archimedean norm on $R$.


Proof

Since $R$ has characteristic $p > 0$, the set:

$\set{n \cdot 1_k: n \in \Z}$

has cardinality $p - 1$.


Therefore:

$\sup \set {\norm{n \cdot 1_R}: n \in \Z} = \max \set{\norm{1 \cdot 1_R}, \norm{2 \cdot 1_R}, \cdots, \norm{\paren{p-1} \cdot 1_R } } \lt +\infty$


By Corollary 2 then:

$\norm{\,\cdot\,}$ is non-Archimedean and $C = 1$

$\blacksquare$