Relative Galois module structure and Steinitz classes of dihedral extensions of degree 8

Let k be a number field and O-k its ring of integers. Let Gamma be the dihedral group of order 8. Let M be a maximal O-k-order in k[Gamma] containing O-k[Gamma] and Tl(M) its class group. We denote by R(M) the set of realizable classes, that is, the set of classes c epsilon El(M) such that there exists a Galois extension N/k at most tamely ramified, with Galois group isomorphic to Gamma and the class of M X-O k[Gamma] O-N equal to c, where ON is the ring of integers of N. In this article we prove that R(M) is a subgroup of Tl(M) provided that k and the fourth cyclotomic field of Q are linearly disjoint, and the class number of k is odd. To this end we will solve an embedding problem connected with Steinitz classes of Galois extensions. In addition, for any k with odd class number, we show that the set of Steinitz classes of tame dihedral extension of k is the full class group of k. (C) 2000 Academic Press.