Generalized Steinberg Relations

We consider a field $F$ and positive integers $n$, $m$, such that $m$ is not divisible by $\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{r}(F)$ and is prime to $n!$. The absolute Galois group $G_F$ acts on the group ${\mathbb{U}}_n(\mathbb{Z}\mathrm{/}m)$ of all $(n+1)\times (n+1)$ unipotent upper-triangular matrices over $\mathbb{Z}\mathrm{/}m$ cyclotomically. Given $0,1\mathrm{\ne }z\in F$ and an arbitrary list $w$ of $n$ Kummer elements $(z{)}_F$, $(1-z{)}_F$ in $H^1(G_F,{\mu }_m)$, we construct in a canonical way a quotient ${\mathbb{U}}_w$ of ${\mathbb{U}}_n(\mathbb{Z}\mathrm{/}m)$ and a cohomology element ${\rho }^z$ in $H^1(G_F,{\mathbb{U}}_w)$ whose projection to the superdiagonal is the prescribed list. This extends results by Wickelgren, and in the case $n=2$ recovers the Steinberg relation in Galois cohomology, proved by Tate.