On theG-matrices with entries and eigenvalues inQ(i)

LetG be a finite abelian group,K a subfield ofC, C[G] regarded as an algebra of matrices.A G K {A?C[G]| all the entries and eigenvalues ofA are inK} is an association algebra overK. In this paper, the association scheme ofA G K is determined and in the caseK=Q(i), the first eigenmatrix of the association scheme computed. As an application, it is proved thatZ 4×Z 4×Z 4 is the only abelian group admitted as a Singer group by some distance-regular digraph of girth 4 on 64 vertices.