Definition:Non-Archimedean/Norm (Division Ring)

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Definition

Let $\struct {R, +, \circ}$ be a division ring whose zero is denoted $0_R$.


Definition 1

A norm $\norm {\, \cdot \,}$ on $R$ is non-Archimedean if and only if $\norm {\, \cdot \,}$ satisfies the axiom:

\((N4)\)   $:$   Ultrametric Inequality:      \(\displaystyle \forall x, y \in R:\)    \(\displaystyle \norm {x + y} \)   \(\displaystyle \le \)   \(\displaystyle \max \set {\norm x, \norm y} \)             


Definition 2

A non-Archimedean norm on $R$ is a mapping from $R$ to the non-negative reals:

$\norm {\,\cdot\,}: R \to \R_{\ge 0}$

satisfying the non-Archimedean norm axioms:

\((N1)\)   $:$   Positive Definiteness:      \(\displaystyle \forall x \in R:\)    \(\displaystyle \norm x = 0 \)   \(\displaystyle \iff \)   \(\displaystyle x = 0_R \)             
\((N2)\)   $:$   Multiplicativity:      \(\displaystyle \forall x, y \in R:\)    \(\displaystyle \norm {x \circ y} \)   \(\displaystyle = \)   \(\displaystyle \norm x \times \norm y \)             
\((N4)\)   $:$   Ultrametric Inequality:      \(\displaystyle \forall x, y \in R:\)    \(\displaystyle \norm {x + y} \)   \(\displaystyle \le \)   \(\displaystyle \max \set{\norm x, \norm y} \)             


The pair $\struct {R, \norm {\, \cdot \, } }$ is a non-Archimedean Normed Division Ring.


If $R$ is also a commutative ring, that is, $\struct {R, \norm {\,\cdot\,} }$ is a valued field, then $\struct {R, \norm {\,\cdot\,} }$ is a non-Archimedean Valued Field.


Archimedean

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


Also see

Sources