On the structure of hyperfields obtained as quotients of fields

TL;DR

This paper classifies hyperfields of finite order that can be formed as quotients of finite fields, using number theory and Ramsey theory, and identifies all such hyperfields of order up to 4.

Contribution

It provides a complete classification of hyperfields obtained as quotients of finite fields of large order, including all hyperfields of order at most 4.

Findings

All hyperfields of a given finite order that are quotients of large finite fields are classified.

Hyperfields of order at most 4 that are quotients of fields are explicitly identified.

Number theory and Ramsey theory are used as key tools in the classification process.

Abstract

We determine all isomorphism classes of hyperfields of a given finite order which can be obtained as quotients of finite fields of sufficiently large order. Using this result, we determine which hyperfields of order at most 4 are quotients of fields. The main ingredients in the proof are the Weil bounds from number theory and a result from Ramsey theory.