Abstract Let p be a prime number and let F be a field of characteristic different from p. We prove that there exist a field extension L/F and a, b, c, d in L^ such that (a, b) = (b, c) = (c, d) =0 in Br (L) p but the mod p Massey product a, b, c, d is not defined over L. Thus, the strong Massey vanishing conjecture at the prime p fails for L, and the cochain differential graded ring C^* (L, Z/p Z) of the absolute Galois group L of L is not formal. This answers a question of Positselski. As our main tool, we define a secondary obstruction that detects non-triviality of unramified torsors under tori, and which is of independent interest.
Merkurjev et al. (Tue,) studied this question.