Definition:Non-Archimedean/Norm (Division Ring)/Definition 1

Definition

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

A norm $\norm {\, \cdot \,}$ on $R$ is non-Archimedean if and only if $\norm {\, \cdot \,}$ satisfies the axiom:

\((\text N 4)\)

$:$

Ultrametric Inequality:

\(\ds \forall x, y \in R:\)

\(\ds \norm {x + y} \)

\(\ds \le \)

\(\ds \max \set {\norm x, \norm y} \)

Also see

• Equivalence of Definitions of Non-Archimedean Division Ring Norm

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.1$: Absolute Values on a Field: Definition $2.1.1 \ \text {(iv)}$
• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$ Normed fields: Proposition $1.12$