Construction of unipotent Galois extensions and Massey products

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 p6. 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.