On the hyperfields associated to valued fields

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.

A Result of Krasner in Categorial Form

“…We now define a projective system of hyperfields associated with any valued field of arbitrary Archimedean rank (see also [9,10]).…”

“…The question whether all hyperfields admitting non-trivial valuations can be obtained from fields with that multiplicative quotient construction is, to our knowledge, open. This particular problem is more extensively discussed in [10].…”

“…A hyperfield is a field-like structure where the latter property is relaxed for the additive operation. In the literature, such structures appear perhaps more than one would expect: hyperfields are of interest, e.g., in tropical geometry [1][2][3], symmetrization [4][5][6], projective geometry [7], valuation theory [8][9][10][11], and ordered algebra [12][13][14]. There are even reasons to believe that their theory generalizes field theory in ways that can be used to tackle deep problems such as the description of F 1 , the "field of characteristic one" (cf.…”

“…A diagram of the form (3) as above can be associated with any non-trivially valued hyperfield (cf. [10] Proposition 1.17). As a corollary to the results presented in this article, we may deduce the following statement:…”

“…Let us note that the hypothesis of the above assertion holds under weaker assumptions on (K, v) than that of K being a field. For instance, by [10] (Proposition 1.27), it suffices that v satisfies Krasner's further assumption (such valuations are called Krasner valuations in [24] (Section 4)). The problem of whether the just mentioned assumption on (K, v) is also necessary is outside the scope of this work and left open for future investigations.…”