Definition:Non-Archimedean/Norm (Division Ring)
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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:
| \((\text N 4)\) | $:$ | Ultrametric Inequality: | \(\ds \forall x, y \in R:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \max \set {\norm x, \norm y} \) |
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:
| \((\text N 1)\) | $:$ | 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: | \(\ds \forall x, y \in R:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \max \set {\norm x, \norm y} \) |
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 $\norm {\, \cdot \,}$ is said to be Archimedean if and only if it does not satisfy the Non-Archimedean Norm Axiom $\text N 4$: Ultrametric Inequality.
Also see
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.1$: Absolute Values on a Field: Definition $2.1.1 \ \text {(iv)}$
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$ Normed fields: Proposition $1.12$