A pr 2 00 5 GALOIS MODULE STRUCTURE OF p TH - POWER CLASSES OF CYCLIC EXTENSIONS OF DEGREE

In the mid-1960s Borevič and Faddeev initiated the study of the Galois module structure of groups of pth-power classes of cyclic extensions K/F of pth-power degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois modules for all base fields F . In 1947 Šafarevič initiated the study of Galois groups of maximal pextensions of fields with the case of local fields [12], and this study has grown into what is both an elegant theory as well as an efficient tool in the arithmetic of fields. From the very beginning it became clear that the groups of pth-power classes of the various field extensions of a base field encode basic information about the structure of the Galois groups of maximal p-extensions. (See [7] and [13].) Such groups of pth-power classes arise naturally in studies in arithmetic algebraic geometry, for example in the study of elliptic curves. In 1960 Faddeev began to study the Galois module structure of pthpower classes of cyclic p-extensions, again in the case of local fields, and during the mid-1960s he and Borevič established the structure of these Galois modules using basic arithmetic invariants attached to Galois extensions. (See [6] and [4].) In 2003 two of the authors ascertained the Galois module structure of pth-power classes in the case of cyclic extensions of degree p over all base fields F containing a primitive pth root of unity [9]. Very recently, this work paved the way for the determination of the entire Galois cohomology as a Galois module in 2000 Mathematics Subject Classification. Primary 12F10; Secondary 16D70.