GALOIS MODULE STRUCTURE OF pTH-POWER CLASSES OF EXTENSIONS OF DEGREE p

For fields F of characteristic not p containing a primitive pth root of unity, we determine the Galois module structure of the group of pth-power classes of K for all cyclic extensions K/F of degree p. The foundation of the study of the maximal p-extensions of fields K containing a primitive pth root of unity is a group of the pth-power classes of the field: by Kummer theory this group describes elementary p-abelian quotients of a maximal p-extension. The size of the group controls the number of generators of the Galois group of the maximal p-extension. Furthermore, as K varies among the Galois extensions of the base field F , the Galois module structure of pth-power classes plays a fundamental role in the investigation of the Galois group of the maximal p-extension of F . In particular, in the nineteen-sixties Borevič and Faddeev closely studied the Galois module structure of the pth-power classes of K for cyclic extensions K/F of degree p in the case when F is a local field, and in the seventies Miki used this structure in the service of Galois embedding problems (see [Bo], [F], and [Mi]). From a cohomological point of view, the Galois module structure of pth-power classes is especially important. These Galois modules may be naturally identified with the first cohomology groups of maximal p-extensions with Fp-coefficients. (See [S, Chapter 1].) It is moreover conjectured that the entire Galois cohomology ring of K with coefficients in Fp of degree at least one is generated by its first cohomology group. (See [V] for a proof in the case of p = 2 as well as comments on this conjecture which was initially considered by Beilinson, BlochKato, Lichtenbaum, and Milnor.) As a result, for the investigation of the action of the Galois group Gal(K/F ) on the Galois cohomology Research supported in part by the Natural Sciences and Engineering Research Council of Canada and by the special Dean of Science Fund at the University of Western Ontario. Supported by the Mathematical Sciences Research Institute, Berkeley.