Realizable Galois module classes for tetrahedral extensions

Abstract

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 Sodaïgui (J. Number Theory98 (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.