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Transactions of the American Mathematical Society

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Galois groups and cohomological functors


Authors: Ido Efrat and Ján Mináč
Journal: Trans. Amer. Math. Soc. 369 (2017), 2697-2720
MSC (2010): Primary 12G05, 12E30
DOI: https://doi.org/10.1090/tran/6724
Published electronically: July 29, 2016
MathSciNet review: 3592525
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Abstract: Let $ q=p^s$ be a prime power, $ F$ a field containing a root of unity of order $ q$, and $ G_F$ its absolute Galois group. We determine a new canonical quotient $ \mathrm {Gal}(F_{(3)}/F)$ of $ G_F$ which encodes the full mod-$ q$ cohomology ring $ H^*(G_F,\mathbb{Z}/q)$ and is minimal with respect to this property. We prove some fundamental structure theorems related to these quotients. In particular, it is shown that when $ q=p$ is an odd prime, $ F_{(3)}$ is the compositum of all Galois extensions $ E$ of $ F$ such that $ \mathrm {Gal}(E/F)$ is isomorphic to $ \{1\}$, $ \mathbb{Z}/p$ or to the nonabelian group $ H_{p^3}$ of order $ p^3$ and exponent $ p$.


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Additional Information

Ido Efrat
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Be’er-Sheva 84105, Israel
Email: efrat@math.bgu.ac.il

Ján Mináč
Affiliation: Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada
Email: minac@uwo.ca

DOI: https://doi.org/10.1090/tran/6724
Received by editor(s): August 12, 2013
Received by editor(s) in revised form: February 15, 2015, and April 18, 2015
Published electronically: July 29, 2016
Additional Notes: The first author was supported by the Israel Science Foundation (grants No. 23/09 and 152/13)
The second author was supported in part by National Sciences and Engineering Council of Canada grant R0370A01.
Article copyright: © Copyright 2016 American Mathematical Society