  Given a finite group G, we say that a subset C of G is power-closed if, for
every x in C and y in <x> with <x>=<y>, we have that y lies in C.
  In this paper we are interested in finite Cayley digraphs Cay(G,C) over G
with connection set C, where C is a union of conjugacy classes of G. We show
that each eigenvalue of Cay(G,C) is integral if and only if C is power-closed.
This result will follow from a discussion of some more general rationality
conditions on the eigenvalues of Cay(G,C).
