  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.
