Normed field

Remarks

Example: $\C$ with the modulus $\abs{a + bi} = \sqrt{a^2 + b^2}$ is a normed field; clause 2 is $\abs{zw} = \abs{z}\abs{w}$, and clause 3 is the triangle inequality in the plane. Example: $\Q$ with the p-adic absolute value for a prime $p$ is a normed field. Example: every field with the trivial absolute value, $\abs{x} = 1$ for $x \ne 0$, is a normed field. Non-example: on $\R$, the map $x \mapsto x^2$ satisfies clauses 1 and 2 but fails clause 3, since $(1 + 1)^2 = 4 > 2 = 1^2 + 1^2$. Clauses 1 and 2 give $\abs{1} = 1$ and $\abs{-x} = \abs{x}$. An absolute value that satisfies the stronger inequality $\abs{x + y} \le \max(\abs{x}, \abs{y})$ is a non-Archimedean absolute value. A normed field is a topological field for the topology of its metric.