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Cohomological Dimension and Schreier's Formula in Galois Cohomology

Published online by Cambridge University Press:  20 November 2018

John Labute ,
Nicole Lemire ,
Ján Mináč  and
John Swallow
John Labute
Affiliation:
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke StreetWest, Montreal, QC, H3A 2K6 e-mail: labute@math.mcgill.ca
Nicole Lemire
Affiliation:
Department of Mathematics, Middlesex College, University of Western Ontario, London, ON, N6A 5B7 e-mail: nlemire@uwo.caminac@uwo.ca
Ján Mináč
Affiliation:
Department of Mathematics, Middlesex College, University of Western Ontario, London, ON, N6A 5B7 e-mail: nlemire@uwo.caminac@uwo.ca
John Swallow
Affiliation:
Department of Mathematics, Davidson College, Davidson, NC 28035-7046, U.S.A. e-mail: joswallow@davidson.edu
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Abstract

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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 ${{\dim}_{{{\mathbb{F}}_{p}}}}{{H}^{n}}\left( H,\ {{\mathbb{F}}_{p}} \right)$.

Keywords

12G0512G10cohomological dimensionSchreier's formulaGalois theoryp-extensionspro-p-groups
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 4 , 01 December 2007 , pp. 588 - 593
Copyright
Copyright © Canadian Mathematical Society 2007

References

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