Extensions of Hyperfields

TL;DR

This paper develops a theory of hyperfield extensions, introducing weak and strong types, and explores their properties, including the construction of extensions with roots and the non-uniqueness of such extensions.

Contribution

It generalizes field extension concepts to hyperfields, providing methods to construct strong extensions and highlighting the non-uniqueness of adjoining roots.

Findings

Constructed strong hyperfield extensions with roots for any polynomial

Identified hyperfields with non-isomorphic minimal extensions containing a root

Showed that adjoining roots is not a well-defined operation in hyperfields

Abstract

We develop a theory of extensions of hyperfields that generalizes the notion of field extensions. Since hyperfields have a multivalued addition, we must consider two kinds of extensions that we call weak hyperfield extensions and strong hyperfield extensions. For quotient hyperfields, we develop a method to construct strong hyperfield extensions that contain roots to any polynomial over the hyperfield. Furthermore, we give an example of a hyperfield that has two non-isomorphic minimal extensions containing a root to some polynomial. This shows that the process of adjoining a root to a hyperfield is not a well-defined operation.