Galois module structure of Galois cohomology

Nicole Lemire, Jan Minac and John Swallow

\newcommand{\F}{\mathbb{F}} \newcommand{\Fp}{\F_p}

Let $F$ be a field containing a primitive $p$th root of unity, and let $U$ be an open normal subgroup of index $p$ of the absolute Galois group $G_F$ of $F$. We determine the structure of the cohomology group $H^n(U,\Fp)$ as an $\Fp[G_F/U]$-module for all $n\in\mathbb{N}$. Previously this structure was known only for $n=1$, and until recently the structure even of $H^1(U,\Fp)$ was determined only for $F$ a local field, a case settled by Borevi\v{c} and Faddeev in the 1960s.