  We prove that all algebraic bases $\beta$ allow an eventually periodic
representations of the elements of $\mathbb Q(\beta)$ with a finite alphabet of
digits $\mathcal A$. Moreover, the classification of bases allowing that those
representations have the so-called weak greedy property is given.
  The decision problem whether a given pair $(\beta,\mathcal A)$ allows
eventually periodic representations proves to be rather hard, for it is
equivalent to a topological property of the attractor of an iterated function
system.
