On the hyperfields associated to valued fields

2022/11/09 by Linzi, Alessandro, Touchard, Pierre · 2 citations
#FOS: Mathematics #Logic (math.LO) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2211.05082

Abstract

One can associate to a valued field an inverse system of valued hyperfields (H_i)_{i ∈ 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 ⊕ and ⋅ and two constant symbols 0 and 1.

Cited by

• Automorphisms of valued fields: amalgamation and existential closedness
• Algorithmic Construction of Real Hyperfields from Minimal Axioms

Related

• The Ax-Kochen-Ershov principles via the higher valued hyperfield
• A result of Krasner in categorial form
• Notes on valuation theory for Krasner hyperfields
• An equivalence between two approaches to limits of local fields
• Orderings and valuations in hyperfields