The class of Krasner hyperfields is not elementary

TL;DR

This paper proves that the class of Krasner hyperfields cannot be characterized by first-order logic, using field extension properties and Chebotarev's density theorem.

Contribution

It demonstrates that Krasner hyperfields form a non-elementary class by analyzing rational ranks of quotient groups in field extensions.

Findings

Krasner hyperfields are not elementary.

Rational rank of quotient groups is key to the proof.

Chebotarev's density theorem is utilized in the argument.

Abstract

We show that the class of Krasner hyperfields is not elementary. To show this, we determine the rational rank of quotients of multiplicative groups in field extensions. Our argument uses Chebotarev's density theorem. We also discuss some related questions.