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.

A major difficult problem in Galois theory is the characterization of profinite groups which are realizable as absolute Galois groups of fields. Recently the Kernel n-Unipotent Conjecture and the Vanishing n-Massey Conjecture for n ≥ 3 were formulated. These conjectures evolved in the last forty years as a byproduct of the application of topological methods to Galois cohomology. We show that both of these conjectures are true for odd rigid fields. This is the first case of a significant family of fields where both of the conjectures are verified besides fields whose Galois groups of p-maximal extensions are free pro-p-groups. We also prove the Kernel Unipotent Conjecture for Demushkin groups of rank 2, and establish various filtration results for free pro-p-groups, provide examples of pro-p-groups which do not have the kernel n-unipotent property, compare various Zassenhaus filtrations with the descending p-central series and establish new type of automatic Galois realization.

We compute the F p -dimension of an n-th graded piece G (n) /G (n+1) of the Zassenhaus filtration for various finitely generated pro-p-groups G. These groups include finitely generated free pro-p-groups, Demushkin pro-p-groups and their free pro-p products. We provide a unifying principle for deriving these dimensions.

For all primes p and for all fields, we find a sufficient and necessary condition of the existence of a unipotent Galois extension of degree p 6 . The main goal of this paper is to describe an explicit construction of such a Galois extension over fields admitting such a Galois extension. This construction is surprising in its simplicity and generality. The problem of finding such a construction has been left open since 2003. Recently a possible solution of this problem gained urgency because of an effort to extend new advances in Galois theory and its relations with Massey products in Galois cohomology.

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Let K be a global field which contains a primitive p-th root of unity, where p is a prime number. M. J. Hopkins and K. G. Wickelgren showed that for p = 2, any triple Massey product over K with respect to F p , contains 0 whenever it is defined. We show that this is true for all primes p.

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