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

Jump to navigation Jump to search

## 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.