On the hyperfields associated to valued fields

Abstract

One can associate to a valued field an inverse system of valued hyperfields $(\mathcal{H}_i)_{i \in I}$ in a natural way. We investigate when, conversely, such a system arise from a valued field. First, we extend a result of Krasner by showing that the inverse limit of certain systems are stringent valued hyperfields. Secondly, we describe a Hahn-like construction which yields a henselian valued field from a stringent valued hyperfield. In addition, we provide an axiomatisation of the theory of stringent valued hyperfields in a language consisting of two binary function symbols $\oplus$ and $\cdot$ and two constant symbols $\textbf{0}$ and $\textbf{1}$.