Characterisation of Non-Archimedean Division Ring Norms/Corollary 4

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
Because $R$ has characteristic $p > 0$, the set:
 * $\set {n \cdot 1_k: n \in \Z}$

has cardinality $p - 1$.

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

By Corollary 2:
 * $\norm{\,\cdot\,}$ is non-Archimedean and $C = 1$.