Definition:Non-Archimedean Norm Axioms

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

Let $\norm {\, \cdot \,}: R \to \R_{\ge 0}$ be a non-Archimedean norm on $R$.

The non-Archimedean norm axioms are the conditions on $\norm {\, \cdot \,}$ which are satisfied for all elements of $R$ in order for $\norm {\, \cdot \,}$ to be a non-Archimedean norm:

\((\text N 1)\)   $:$   Positive Definiteness:      \(\displaystyle \forall x \in R:\)    \(\displaystyle \norm x = 0 \)   \(\displaystyle \iff \)   \(\displaystyle x = 0_R \)             
\((\text N 2)\)   $:$   Multiplicativity:      \(\displaystyle \forall x, y \in R:\)    \(\displaystyle \norm {x \circ y} \)   \(\displaystyle = \)   \(\displaystyle \norm x \times \norm y \)             
\((\text N 4)\)   $:$   Ultrametric Inequality:      \(\displaystyle \forall x, y \in R:\)    \(\displaystyle \norm {x + y} \)   \(\displaystyle \le \)   \(\displaystyle \max \set {\norm x, \norm y} \)             

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