  A graph is Cartesian decomposable if it is isomorphic to a Cartesian product
of (more than one) strictly smaller graphs, each of which has more than one
vertex and admits no such decomposition. These smaller graphs are called the
Cartesian-prime factors of the Cartesian decomposition, and were shown, by
Sabidussi and Vizing independently, to be uniquely determined up to
isomorphism. We characterise by their parameters those generalised Paley graphs
which are Cartesian decomposable, and we prove that for such graphs, the
Cartesian-prime factors are themselves smaller generalised Paley graphs. This
generalises a result of Lim and the second author which deals with the case
where all the Cartesian-prime factors are complete graphs. These results
contribute to the determination, by parameters, of generalised Paley graphs
with automorphism groups larger than the 1-dimensional affine subgroups used to
define them.
