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.

Classes réalisables d'extensions métacycliques de degré lm

“…Pour des résultats récents dans la direction de l'étude de la conjecture non abélienne sur les classes réalisables voir [2][3][4][5]22].…”

Classes réalisables d'extensions non abéliennes de degré p3

“…Pour des résultats récents dans la direction de l'étude de la conjecture non abélienne sur les classes réalisables voir [3][4][5][6][7]14,19].…”