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

< Definition:Non-Archimedean(Redirected from Definition:Non-Archimedean Division Ring Norm)

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## 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.

## Also see

## Sources

- 1997: Fernando Q. Gouvea:
*p-adic Numbers: An Introduction*... (previous) ... (next): $\S 2.1$: Absolute Values on a Field, Definition $2.1.1 \, (iv)$

- 2007: Svetlana Katok:
*p-adic Analysis Compared with Real*... (previous) ... (next): $\S 1.2$ Normed fields, Proposition $1.12$