Galois module structure of elementary abelian extensions

Abstract

Let K be an algebraic number field, ο=O_K its ring of integers, and G an elementary abelian group of order l_k. In this article, we determine which classes in the locally free class group Cl(οG) of the group ring οG are realizable as Galois module classes of rings of integers O_L in tame Galois extensions $\text{L}\text{K}$ with $\text{Gal}\text{(}\text{L}\text{K}\text{) ≅ G}$. The set R(οG) of such realizable classes is described in terms of the action on Cl(οG) of a Stickelberger ideal $\text{J}$ in the integral group ring $\text{Z}$C, where C (≅$\text{F}$^x_lk) is a (Cartan) subgroup of the automorphism group Aut G (≅GL_k($\text{F}$_l)).