On the structure of hyperfields obtained as quotients of fields

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.

9 pages. v2: fixed errors and improved exposition following referee report. To appear in Proceedings of the AMS