Axiom:Non-Archimedean Norm Axioms/Division Ring

Definition

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

Let $\norm {\, \cdot \,}: R \to \R_{\ge 0}$ be a non-Archimedean norm on $R$.

The non-Archimedean norm axioms are the conditions on $\norm {\, \cdot \,}$ which are satisfied for all elements of $R$ in order for $\norm {\, \cdot \,}$ to be a non-Archimedean norm:

\((\text N 1)\)	$:$		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

• Definition:Non-Archimedean Division Ring Norm

• Axiom:Submultiplicative Norm Axioms