Abstract. 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 1947Šafarevič 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

Let K/F be a cyclic field extension of odd prime degree. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F ) such that corresponding normal subgroups are indecomposable F p [Gal(K/F )]-modules. For these embedding problems we prove conditions on solvability, formulas for explicit construction, and results on automatic realizability.

Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group G F of F . Using the Bloch-Kato Conjecture we determine the structure of the cohomology group H n (U, F p ) as an F p [G F /U ]-module for all n ∈ N. Previously this structure was known only for n = 1, and until recently the structure even of H 1 (U, F p ) was determined only for F a local field, a case settled by Borevič and Faddeev in the 1960s. For the case when the maximal pro-p quotient T of G F is finitely generated, we apply these results to study the partial Euler-Poincaré characteristics of χ n (N ) of open subgroups N of T . We show in particular that the nth partial Euler-Poincaré characteristic χ n (N ) is determined by only χ n (T ) and the conorm in H n (T, F p ).

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