# Valued Fields and Normed Spaces

Chapter
Part of the Progress in Mathematics book series (PM, volume 218)

## Abstract

A valued field is a field k provided with a valuation (or absolute value). The latter is a map a ↦ |a| from k to R which satisfies the rules:
$$\begin{gathered} 1. \left| a \right| \geqslant 0 and \left| a \right| = 0 if and only if a = 0. \hfill \\ 2. \left| {ab} \right| = \left| a \right| \cdot \left| b \right|. \hfill \\ 3. \left| {a + b} \right| \leqslant \left| a \right| + \left| b \right|. \hfill \\ \end{gathered}$$
The absolute values on the fields R and C make these fields into valued fields. In the present context the most important valued fields are the so-called nonarchimedean valued fields. They are defined by replacing the triangle inequality $$\left| {a + b} \right| \leqslant \left| a \right| + \left| b \right|$$ with the stronger inequality $$\left| {a + b} \right| \leqslant \max \left( {\left| a \right|,\left| b \right|} \right)$$. The trivial valuation is defined by $$\left| 0 \right| = 0 and \left| a \right| = 1 for a \in k^*$$. In what follows we will mean by valuation and valued field, a non-trivial, non-archimedean valuation and a field equipped with such a valuation.