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# Cohomological Dimension and Schreier's Formula in Galois Cohomology

Published:2007-12-01
Printed: Dec 2007
• John Labute
• Nicole Lemire
• Ján Mináč
• John Swallow
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## Abstract

Let $p$ be a prime and $F$ a field containing a primitive $p$-th root of unity. Then for $n\in \N$, the cohomological dimension of the maximal pro-$p$-quotient $G$ of the absolute Galois group of $F$ is at most $n$ 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 generalize Schreier's formula for $\dim_{\Fp} H^1(H,\Fp)$ to $\dim_{\Fp} H^n(H,\Fp)$.
 Keywords: cohomological dimension, Schreier's formula, Galois theory, $p$-extensions, pro-$p$-groups
 MSC Classifications: 12G05 - Galois cohomology [See also 14F22, 16Hxx, 16K50] 12G10 - Cohomological dimension

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