Definition:Trivial Norm/Division Ring

Definition

Let $\struct {R, +, \circ}$ be a division ring, and denote its zero by $0_R$.

Then the map $\norm {\cdot}: R \to \R_{\ge 0}$ given by:

$\norm x = \begin{cases}

 0 & : \text{if $x = 0_R$}\\
 1 & : \text{otherwise}

\end{cases}$

defines a norm on $R$, called the trivial norm.

Also known as

Some authors refer to this norm as the trivial absolute value.

Nontrivial

A norm $\norm {\, \cdot \,}$ on a division ring $R$ is nontrivial if and only if it is not trivial.

Also see

• Trivial Norm on Division Ring is Norm

• Definition:Standard Discrete Metric

• Nontrivial Norm on Division Ring

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S2.1$ Absolute Values on a Field, Definition $2.1.1$, Example $2$.

• 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S1.2$ Normed Fields, Definition $1.5$