Valued Fields and Normed Spaces

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


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.


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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  1. 1.Théorie des nombres et Algorithmique arithmétique (A2X) UMR 5465Université Bordeaux 1TalenceFrance
  2. 2.Department of MathematicsUniversity of GroningenGroningenThe Netherlands

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