We show that any triple Massey product with respect to prime 2 contains 0 whenever it is defined over any field. This extends the theorem of M. J. Hopkins and K. G. Wickelgren, from global fields to any fields. This is the first time when the vanishing of any n-Massey product for some prime p has been established for all fields. This leads to a strong restriction on the shape of relations in the maximal pro-2-quotients of absolute Galois groups, which was out of reach until now. We also develop an extension of Serre's transgression method to detect triple commutators in relations of pro-p-groups, where we do not require that all cup products vanish. We prove that all n-Massey products, n ≥ 3, vanish for general Demushkin groups. We formulate and provide evidence for two conjectures related to the structure of absolute Galois groups of fields. In each case when these conjectures can be verified, they have some interesting concrete Galois theoretic consequences. They are also related to the Bloch-Kato conjecture.
ABSTRACT. We show that the absolute Galois group of any field has the vanishing triple Massey product property. Several corollaries for the structure of maximal pro-p-quotient of absolute Galois groups are deduced. Furthermore, the vanishing of some higher Massey products is proved.
Let p be an odd prime number and F a field containing a primitive pth root of unity. We prove a new restriction on the group-theoretic structure of the absolute Galois group G
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
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.