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

Abstract

Let k be a number field with ring of integers O k , 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 O N of N determines a class in the locally free class group Cl ( O k [ Γ ] ) . We show that the set of classes in Cl ( O k [ Γ ] ) realized in this way is the kernel of the augmentation homomorphism from Cl ( O k [ Γ ] ) to the ideal class group Cl ( O k ) , provided that the ray class group of O k for the modulus 4 O k 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 ] .