Periodic elements and number systems in Q(√2)

Let us consider an arbitrary quadratic extension of the field of rational numbers. Our prospective purpose is to give for an arbitrary algebraic integer α, if any, such a digit set that constitutes a number system with α. In this paper, we deal with the periodic elements of systems given in Q(√2) and prove that either the modulus of them or that of their conjugate is less than 1. On the basis of this result, we hope that there exists some algorithm which provides number systems.