Computing all power integral bases of cubic fields

Applying Baker's effective method and the reduction procedure of Baker and Davenport, we present several lists of solutions of index form equations in (totally real and complex) cubic algebraic number fields. These solutions yield all power integral bases of these fields. 1. Introduction. Let K be a cubic algebraic number field and denote by ZK the ring of integers of K. A power integral basis of K is an integral basis of the form {1, &, &2} with some a E ZK. If there exists such an a, then we say that K is monogenic, since ZK = Z(&e). Obviously, if a has this property, then it holds also for a + k with any k E Z. From a practical point of view, it is important to know whether there exists a power integral basis of K, and if so, what are the numerical values of a. If the Galois group of K is cyclic, then the discriminant of K is a full square. The problem of monogeneity in cyclic cubic fields was considered by M. N. Gras (12), (13), Archinard (1) and Dummit and Kisilevsky (4). M. N. Gras and Archinard gave necessary and sufficient conditions for monogeneity and tested for several numerical examples whether or not the field is monogenic. Moreover, Dummit and Kisilevsky proved that there exist infinitely many cyclic cubic fields with power integral bases. For arbitrary algebraic number fields L, it was proved by Gyory (14) that up to obvious translations by elements of Z, there are only finitely many a E ZL with ZL = Z(&), and he gave effective (but rather large) bounds for the sizes of these a. In this paper we present a method which allows us to determine all possible values of a (up to translation with rational integers) such that {1, &, &2} is a power integral