Definition:Non-Archimedean/Norm (Division Ring)/Definition 2

Definition

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

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

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 $1.12$