Realizable classes of tetrahedral extensions

Let k be a number field and O-k its ring of integers. Let Gamma be the alternating group A(4). Let M be a maximal O-k-order in k[Gamma] containing O-k[Gamma] and Cl(M) its class group. We denote by R(M) the set of realizable classes, that is the set of classes c is an element of Cl(M) such that there exists a Galois extension N/k at most tamely ramified, with Galois group isomorphic to Gamma, for which the class of M x O-k[Gamma] O-N is equal to c, where O-N is the ring of integers of N. In this article we determine R(M) and we prove that it is a subgroup of Cl(M) provided that k and the 3rd cyclotomic field of Q are linearly disjoint, and the class number of k is odd. (C) 2002 Elsevier Science (USA). All rights reserved.