Extensions of Hyperfields

2019/12/11 by Steven Creech, Creech, Steven
Mathematics · #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA) #math.AC #math.RA

paper · pdf · doi:10.48550/arxiv.1912.05919

arxiv created 2019/12/11 · arxiv updated 2019/12/13

Abstract

We develop a theory of extensions of hyperfields that generalizes the notion of field extensions. Since hyperfields have a multivalued addition, we must consider two kinds of extensions that we call weak hyperfield extensions and strong hyperfield extensions. For quotient hyperfields, we develop a method to construct strong hyperfield extensions that contain roots to any polynomial over the hyperfield. Furthermore, we give an example of a hyperfield that has two non-isomorphic minimal extensions containing a root to some polynomial. This shows that the process of adjoining a root to a hyperfield is not a well-defined operation.

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