On Z-Modules of Algebraic Integers

Let q be an algebraic integer of degree d >= 2. Consider the rank of the multiplicative subgroup of C* generated by the conjugates of q. We say q is of full rank if either the rank is d - 1 and q has norm +/- 1, or the rank is d. In this paper we study some properties of Z[q] where q is an algebraic integer of full rank. The special cases of when q is a Pisot number and when q is a Pisot-cyclotomic number are also studied. There are four main results. (1) If q is an algebraic interger of full rank and n is a fixed positive integer, then there are only finitely many m such that disc Z[q(m)] - disc Z[q(n)]. (2) If q and r are algebraic integers of degree d of full rank and Z[q(n)] = Z[r(n)] for infinitely many n, then either q omega r' or q = Norm(r)(2/d)omega/r', where r' is some conjugate of r and omega is some root of unity. (3) Let r be an algebraic integer of degree at most 3. Then there are at most 40 Pisot numbers q such that Z[q] - Z[r]. (4) There are only finitely many Pisot-cyclotomic numbers of any fixed order.