John Labute, Nicole Lemire, Jan Minac, John Swallow: Cohomological dimension and Schreier's formula in Galois cohomology

labute@math.mcgill.ca, nlemire@uwo.ca, minac@uwo.ca, joswallow@davidson.edu

Submission: 2004, Nov 20

Let $p$ be a prime and $F$ a field containing a primitive $p$th root of unity. If $p>2$ 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 $\dim_{\Fp}H^1(H,\Fp)$.

2000 Mathematics Subject Classification: 12G05 12G10

Keywords and Phrases: cohomological dimension, Schreier's formula, Galois theory, $p$-extensions, pro-p-groups

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