On the hyperfields associated to valued fields

TL;DR

This paper explores the relationship between valued fields and associated hyperfields, extending known results, constructing henselian fields from hyperfields, and axiomatizing their theory.

Contribution

It extends Krasner's result on inverse limits, introduces a Hahn-like construction for henselian fields, and provides an axiomatization of stringent valued hyperfields.

Findings

Inverse limits of certain systems are stringent valued hyperfields.

A Hahn-like construction yields henselian valued fields from hyperfields.

The theory of stringent valued hyperfields is axiomatized in a specific language.

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 $\textbf{1}$.