Systems of Fundamental Units in Cubic Orders Original Research Article

Let φ(x) = x3 + ex2 + ƒx + g be an irreducible polynomial with three real roots. Let Mλ be the cubic order Z[λ] where φ(λ) = 0. Many techniques in computational number theory depend on a knowledge of fundamental units for the orders being studied. When these fundamental units are of a form that is consistent throughout an entire family of orders, the techniques can often be applied to all of the orders at once. This paper provides a technique for determining fundamental systems of units in families of cubic orders of the form Mλ. The main theorem yields fundamental systems of units for a wide range of cubic orders with only a small number of restrictions. As an application, the main theorem is used to determine new families of orders with fundamental units of a consistent form.