Abstract Let k be a number field and O k its ring of integers. Let Γ be a finite group, N/k a Galois extension with Galois group isomorphic to Γ, and O N the ring of integers of N. Let ℳ be a maximal O k -order in the semi-simple algebra k[Γ] containing O k [Γ], and Cl(ℳ) its locally free classgroup. When N/k is tame (i.e., at most tamely ramified), extension of scalars allows us to assign to O N the class of , denoted , 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 . In the present article, we prove, by means of a fairly explicit description, that ℛ(ℳ) is a subgroup of Cl(ℳ) when , where V is an 𝔽2-vector space of dimension r ≧ 2, C a cyclic group of order 2 r − 1, and p a faithful representation of C in V; an example is the alternating group A 4. In the proof, we use some properties of the binary Hamming code and solve an embedding problem connected with Steinitz classes. In addition, we determine the set of Steinitz classes of tame Galois extensions of k, with the above group as Galois group, and prove that it is a subgroup of the classgroup of k.

Classes réalisables d'extensions non abéliennes

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

On realizable Galois module classes and Steinitz classes of nonabelian extensions

Let k be a number field with ring of integers $\mathfrak{O}_k$ , and let $\Gamma=A_4$ be the tetrahedral group. For each tame Galois extension N / k with group isomorphic to $\Gamma$ , the ring of integers $\mathfrak{O}_{N}$ of N determines a class in the locally free class group $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ . We show that the set of classes in $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ realized in this way is the kernel of the augmentation homomorphism from $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ to the ideal class group $\mathrm{Cl}(\mathfrak{O}_k)$ . This refines a result of Godin and Sodaigui ( J. Number Theory 98 (2003), 320–328) on Galois module structure over a maximal order in $k[\Gamma]$ . To the best of our knowledge, our result gives the first case where the set of realizable classes in $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ has been determined for a nonabelian group $\Gamma$ and an arbitrary number field k .

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Realizable Galois module classes for tetrahedral extensions

Let p be an odd prime number, r a natural number, and H4pr the generalized quaternion group of order 4p r . Let k be a number field and Cl(k) its class group. Let Rm(k,H4pr) be the subset of Cl(k) consisting of those classes which are realizable as Steinitz classes of tame Galois extensions of k with Galois group isomorphic to H4pr. In this article, we determine Rm(k,H4pr) and show that it is a subgroup of Cl(k). In particular, Rm(k,H4pr) is the full group Cl(k) if p does not divide the class number of k, or if either k contains a primitive p r th root of unity or p is unramified in k.

M. Godin

Papers

Realizable classes of tetrahedral extensions