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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Abelian-by-central Galois groups of fields I: A formal description
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by Adam Topaz PDF
Trans. Amer. Math. Soc. 369 (2017), 2721-2745 Request permission


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 $\operatorname {Gal}(K)^{(2)}/\operatorname {Gal}(K)^{(3)} \hookrightarrow \operatorname {Fun}(K\smallsetminus \{0,1\},(\mathbb {Z}/n)^2)$, 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
  • MR Author ID: 1051144
  • Email:
  • 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.
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 2721-2745
  • MSC (2010): Primary 12G05, 12F10, 20J06, 20E18
  • DOI:
  • MathSciNet review: 3592526