  We determine and explicitly parametrize the isomorphism classes of
nonassociative quaternion algebras over a field of characteristic different
from two, as well as the isomorphism classes of nonassociative cyclic algebras
of odd prime degree when the base field contains a primitive $m$th root of
unity. In the course of doing so, we prove that any two such algebras can be
isomorphic only if the cyclic field extension and the chosen generator of the
Galois group are the same. As an application, we give a parametrization of
nonassociative cyclic algebras of prime degree over a local nonarchimedean
field $F$, which is entirely explicit under mild hypotheses on the residual
characteristic. In particular, this gives a rich understanding of the important
class of nonassociative quaternion algebras up to isomorphism over
nonarchimedean local fields.
