Relative Galois module structure of octahedral extensions

Abstract

Let k be a number field, O k its ring of integers and Cl ( k ) its class group. Let Γ be the symmetric (octahedral) group S 4 . Let M be a maximal O k -order in the semisimple algebra k [ Γ ] containing O k [ Γ ] , Cl ( M ) its locally free class group, and Cl ○ ( M ) the kernel of the morphism Cl ( M ) → Cl ( k ) induced by the augmentation M → O k . Let N / k be a Galois extension with Galois group isomorphic to Γ, and O N the ring of integers of N. When N / k is tame (i.e., at most tamely ramified), extension of scalars allows us to assign to O N the class of M ⊗ O k [ Γ ] O N , denoted [ M ⊗ O k [ Γ ] O N ] , in Cl ( M ) . We define the set R ( M ) of realizable classes to be the set of classes c ∈ Cl ( M ) such that there exists a Galois extension N / k which is tame, with Galois group isomorphic to Γ, and for which [ M ⊗ O k [ Γ ] O N ] = c . In the present article, we prove that R ( M ) is the subgroup Cl ○ ( M ) of Cl ( M ) provided that the class number of k is odd.