On Characteristics of Hyperfields Obtained as Quotients of Finite Fields

TL;DR

This paper investigates the characteristics of hyperfields formed as quotients of finite fields by subgroups of units, providing explicit formulas for certain cases and a general form for prime subgroup orders.

Contribution

It introduces explicit formulas for the characteristic of hyperfields $_p/G$ for specific subgroup sizes and generalizes the characteristic determination for prime subgroup orders.

Findings

Explicit characteristic formulas for $|G|=1,2,3,4$

General form of characteristic for prime $|G|$

Extension of Krasner's hyperfield quotient results

Abstract

Hyperstructures are a natural extension of regular algebraic structures in which one of the operations, known as the hyperoperation, is multivalued; a hyperfield is such an extension on a field. M. Krasner (1962) proved that the quotient $\mathbb{F}_p/G$, where $G$ is a subgroup of units in $\mathbb{F}_p$ is a hyperfield. The characteristic of a field may be explicitly determined from the order of the field, but there are no existing generalizations for determining the characteristic of a hyperfield of the form $\mathbb{F}_p/G$. We show that for odd primes $p$, there exists an explicit form for the characteristic of the hyperfield $\mathbb{F}_p/G$ and $|G|=1,2,3,4$. Finally, we prove a general form of the characteristic for hyperfields where $|G|$ is prime.