Elements of small norm in Shanks' cubic extensions of imaginary quadratic fields

Let k = Q(√−D) be an imaginary quadratic number field with ring of integers Zk and let k(α) be the cubic extension of k generated by the polynomial ft (x) = x3 − (t − 1)x2 − (t + 2)x − 1 with t ∈ Zk . In the present paper we characterize all elements γ ∈ Zk [α] with norms satisfying |Nk(α)/k | ≤ |2t + 1| for |t | ≥ 14. This generalizes a corresponding result by Lemmermeyer and Pethő for Shanks’ cubic fields over the rationals. © 2004 Elsevier Ltd. All rights reserved.