Definition:Equivalent Division Ring Norms/Norm is Power of Other Norm

Definition

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 if and only if $\exists \alpha \in \R_{\gt 0}: \forall x \in R: \norm{x}_1 = \norm{x}_2^\alpha$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.1$ Absolute Values on $\Q$: Lemma $3.1.2$ and Problem $66$

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$ Normed Fields: Proposition $1.10$ and Exercise $13$