Extensions of Hyperfields
Rings and Algebras
2019-12-13 v1 Commutative Algebra
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.
Keywords
Cite
@article{arxiv.1912.05919,
title = {Extensions of Hyperfields},
author = {Steven Creech},
journal= {arXiv preprint arXiv:1912.05919},
year = {2019}
}