\begin{definition}[Definition:Equivalent Division Ring Norms/Norm is Power of Other Norm]
Let $R$ be a division ring.
Let $\norm{\,\cdot\,}_1: R \to \R_{\ge 0}$ and $\norm{\,\cdot\,}_2: R \to \R_{\ge 0}$ be norms on $R$.
$\norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ are '''equivalent''' {{iff}} $\exists \alpha \in \R_{\gt 0}: \forall x \in R: \norm{x}_1 = \norm{x}_2^\alpha$
\end{definition}