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Abelian-by-central Galois groups of fields I: A formal description


Author: Adam Topaz
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 12G05, 12F10, 20J06, 20E18
Published electronically: September 1, 2016
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Abstract: Let $ K$ be a field whose characteristic is prime to a fixed positive integer $ n$ such that $ \mu _n \subset K$, and choose $ \omega \in \mu _n$ as a primitive $ n$-th
root of unity. Denote the absolute Galois group of $ K$ by $ \operatorname {Gal}(K)$, and the
mod-$ n$ central-descending series of $ \operatorname {Gal}(K)$ by $ \operatorname {Gal}(K)^{(i)}$. Recall that
Kummer theory, together with our choice of $ \omega $, provides a functorial isomorphism between $ \operatorname {Gal}(K)/\operatorname {Gal}(K)^{(2)}$ and $ \operatorname {Hom}(K^\times ,\mathbb{Z}/n)$. Analogously to Kummer theory, in this note we use the Merkurjev-Suslin theorem to construct a continuous, functorial and explicit embedding $ \operatorname {Gal}(K)^{(2)}/\operatorname {Gal}(K)^{(3)} \hookrightarrow \linebreak\operatorname {Fun}(K\smallsetminus \{0,1\},(\mathbb{Z}/n)^2)$, where $ \operatorname {Fun}(K\smallsetminus \{0,1\},(\mathbb{Z}/n)^2)$ denotes the group of $ (\mathbb{Z}/n)^2$-valued functions on $ K\smallsetminus \{0,1\}$. We explicitly determine the functions associated to the image of commutators and $ n$-th powers of elements of $ \operatorname {Gal}(K)$ under this embedding. We then apply this theory to prove some new results concerning relations between elements in abelian-by-central Galois groups.


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Additional Information

Adam Topaz
Affiliation: Department of Mathematics, 970 Evans Hall #3840, University of California, Berkeley, Berkeley, California 94720-3840
Email: atopaz@math.berkeley.edu

DOI: https://doi.org/10.1090/tran/6740
Keywords: Abelian-by-central, Galois groups, group cohomology, profinite groups
Received by editor(s): August 26, 2014
Received by editor(s) in revised form: April 20, 2015
Published electronically: September 1, 2016
Additional Notes: This research was supported by NSF postdoctoral fellowship DMS-1304114.
Article copyright: © Copyright 2016 American Mathematical Society