  As is the case for many curved exponential families, the computation of
maximum likelihood estimates in a multivariate normal model with a Kronecker
covariance structure is typically carried out with an iterative algorithm,
specifically, a block-coordinate ascent algorithm. In this article we highlight
a setting, specified by a coprime relationship between the sample size and
dimension of the Kronecker factors, where the likelihood equations have
algebraic degree one and an explicit, easy-to-evaluate rational formula for the
maximum likelihood estimator can be found. A partial converse of this result is
provided that shows that outside of the aforementioned special setting and for
large sample sizes, examples of data sets can be constructed for which the
degree of the likelihood equations is larger than one.
