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Cohomological Dimension and Schreier's Formula in Galois Cohomology
Published online by Cambridge University Press: 20 November 2018
Abstract
Let $p$ be a prime and $F$ a field containing a primitive $p$-th root of unity. Then for $n\,\in \,\mathbb{N}$, 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 ${{H}^{n}}\left( H,\ {{\mathbb{F}}_{p}} \right)\,\to \,{{H}^{n}}\left( G,\ {{\mathbb{F}}_{p}} \right)$ are surjective for all open subgroups $H$ of index $p$. Using this result, we generalize Schreier's formula for ${{\dim}_{{{\mathbb{F}}_{p}}}}\,{{H}^{1}}\,\left( H,\ {{\mathbb{F}}_{p}} \right)$ to ${{\dim}_{{{\mathbb{F}}_{p}}}}{{H}^{n}}\left( H,\ {{\mathbb{F}}_{p}} \right)$.
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- Copyright © Canadian Mathematical Society 2007
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