Articulo
Summary on non-Archimedean valued fields
Fecha
2018Institución
Resumen
This article summarizes the main properties of ultrametric spaces, valued fields, ordered fields and fields with valuations of higher rank, highlighting their similarities and differences. The most used non-Archimedean valued fields are reviewed, like a completion in the case of the p-adic numbers fields and the Levi-Civita fields, or like an algebraic closure as is sometimes the case of the Puiseux series fields. Also, a study of spherically complete valued fields is presented and their characterization as maximally complete fields is given where the Hahn fields and the Mal’cev-Neumann fields (their p-adic analogues) play an important role as “spherical completions”. Finally several of the metric, topological, algebraic and order properties of the most used non-Archimedean valued fields are collected in a catalog where we can appreciate the analogy between fields that have the same characteristic as their residue class fields and fields that do not satisfy this property.