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arXiv · 1810.04035

On Characteristics of Hyperfields Obtained as Quotients of Finite Fields

Antonio Frigo·Hahn Lheem·Dylan Liu
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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.

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Antonio Frigo, Hahn Lheem, Dylan Liu. 2018-10-07. On Characteristics of Hyperfields Obtained as Quotients of Finite Fields. https://arxiv.org/abs/1810.04035

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