Cohomological dimension and Schreier's formula in Galois cohomology
John Labute, Nicole Lemire, Jan Minac and John Swallow
$ \newcommand{\F}{\mathbb{F}} \newcommand{\Fp}{\F_p} \newcommand{\N}{\mathbb{N}} $
Let $p$ be a prime and $F$ a field containing a primitive $p$th root of unity. If $p \gt 2$ assume also that $F$ is perfect. Then for $n\in \N$, the cohomological dimension of the maximal pro-$p$-quotient $G$ of the absolute Galois group of $F$ is $n$ if and only if the corestriction maps $H^n(H,\Fp) \to H^n(G,\Fp)$ are surjective for all open subgroups $H$ of index $p$. Using this result we derive a surprising generalization to $\dim_{\Fp} H^n(H,\Fp)$ of Schreier's formula for $\dim_{\Fp}H^1(H,\Fp)$.