Relative Galois Module Structure and Steinitz Classes of Dihedral Extensions of Degree 8

Let k be a number field and Ok its ring of integers. Let Γ be the dihedral group of order 8. Let be a maximal Ok-order in k[Γ] containing Ok[Γ] and ℓ( ) its class group. We denote by ( ) the set of realizable classes, that is, the set of classes c ℓ( ) such that there exists a Galois extension N/k at most tamely ramified, with Galois group isomorphic to Γ and the class of Ok[Γ]ON equal to c, where ON is the ring of integers of N. In this article we prove that ( ) is a subgroup of ℓ( ) provided that k and the fourth cyclotomic field of 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.