# Definition:Non-Archimedean Norm Axioms

## Definition

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:

 $(N1)$ $:$ 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: $\displaystyle \forall x, y \in R:$ $\displaystyle \norm {x + y}$ $\displaystyle \le$ $\displaystyle \max \set{\norm x, \norm y}$