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 $L^{ \times}$ such that $(a,b)=(b,c)=(c,d)=0$ in $\mathrm{ Br}(F)[p]$ but $\langle a,b,c,d\rangle$ is not defined over $L$. Thus the Strong Massey Vanishing Conjecture at the prime $p$ fails for $L$, and the cochain differential graded ring $C^*(\Gamma_L,\mathbb{ Z}/p\mathbb{ Z})$ of the absolute Galois group $\Gamma_L$ of $L$ is not formal. This answers a question of Positselski.
Categories: math.NT math.AG
