Galois module structure for dihedral extensions of degree 8: Realizable classes over the group ring

Let k be a number field with ring of integers , and let Γ be the dihedral group of order 8. For each tame Galois extension N/k with group isomorphic to Γ, the ring of integers of N determines a class in the locally free class group . We show that the set of classes in realized in this way is the kernel of the augmentation homomorphism from to the ideal class group , provided that the ray class group of for the modulus has odd order. This refines a result of the second-named author (J. Algebra 223 (2000) 367–378) on Galois module structure over a maximal order in k[Γ].