  The first step in investigating colour symmetries for periodic and aperiodic
systems is the determination of all colouring schemes that are compatible with
the symmetry group of the underlying structure, or with a subgroup of it. For
an important class of colourings of planar structures, this mainly
combinatorial question can be addressed with methods of algebraic number
theory. We present the corresponding results for all planar modules with N-fold
symmetry that emerge as the rings of integers in cyclotomic fields with class
number one. The counting functions are multiplicative and can be encapsulated
in Dirichlet series generating functions, which turn out to be the Dedekind
zeta functions of the corresponding cyclotomic fields.
