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

## 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:

 $(\text N 4)$ $:$ Ultrametric Inequality: $\ds \forall x, y \in R:$ $\ds \norm {x + y}$ $\ds \le$ $\ds \max \set {\norm x, \norm y}$

### 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:

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

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