  For all primes $p$ and 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.
