Construction of unipotent Galois extensions and Massey products

Abstract

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.

Triple Massey products vanish over all fields

“…Some results of this paper (for example, the results on Heisenberg extensions in Subsection 3.2, Lemma 4.2) have already been used in the construction of important Galois groups. Namely, in [20] we succeeded in extending the crucial ideas in this paper together with further ideas in Galois theory to find explicit constructions of Galois extensions L/F with Gal(L/F ) U 4 (F p ) for all fields F and all primes p. For example, Theorem 4.12 and its proof play an important role in finding a crucial submodule of the…”

Redei reciprocity, governing fields and negative Pell

“…D, the map γ embeds C/2C as the 'sum-zero-hyperplane' in F t 2 . Equivalently, one can formulate this as in (12) by saying that the subgroup C [2] ⊂ C of quadratic characters on C is generated by the t characters χ p * i , subject to the relation that their sum…”

“…We can restrict α in (12) to the kernel of the 4-rank map R 4 from ( 15) and compose with ϕ 8 to obtain an F 2 -linear map…”

Extensions of unipotent groups, Massey products and Galois theory

“…Here, let us simply state that the conjecture is known to hold for n = 3 (all fields, all p), for local fields (all n, all p), while [GMT], together with its appendix by Wittenberg, settled the case n = 4, p = 2, when F is a number field. On top of the references above, the reader may consult [HW15], [EM], [MT17a].…”