On Characteristics of Hyperfields Obtained as Quotients of Finite Fields

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.

15 pages, PROMYS Research