Definition:Non-Archimedean/Norm (Division Ring)/Archimedean

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Definition

Let $\norm {\, \cdot \,}$ be a norm on a division ring $R$ satisfying the norm axioms:

\((\text N 1)\)   $:$   Positive Definiteness:      \(\ds \forall x \in R:\)    \(\ds \norm x = 0 \)   \(\ds \iff \)   \(\ds x = 0_R \)      
\((\text N 2)\)   $:$   Multiplicativity:      \(\ds \forall x, y \in R:\)    \(\ds \norm {x \circ y} \)   \(\ds = \)   \(\ds \norm x \times \norm y \)      
\((\text N 3)\)   $:$   Triangle Inequality:      \(\ds \forall x, y \in R:\)    \(\ds \norm {x + y} \)   \(\ds \le \)   \(\ds \norm x + \norm y \)      


A norm $\norm {\, \cdot \,}$ is said to be Archimedean if and only if it does not satisfy the Non-Archimedean Norm Axiom $\text N 4$: Ultrametric Inequality.


That is, $\norm {\, \cdot \,}$ is Archimedean if and only if:

\((\text N 5)\)   $:$     \(\ds \exists x, y \in R:\)    \(\ds \norm {x + y} \)   \(\ds > \)   \(\ds \max \set {\norm x, \norm y} \)      


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