On the hyperfields associated to valued fields

Abstract

One can associate an inverse system of valued hyperfields ${\left(H_i\right)}_{i\in I}$ to a valued field in a natural way. We investigate when, conversely, such a system arises 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, ⊕ and ⋅, and two constant symbols, 0 and 1.