The Kummerian property and maximal pro-\(p\) Galois groups
type: Item
instance of: scholarly article
title: The Kummerian property and maximal pro-\(p\) Galois groups
zbMATH Open document ID: 1446.12005
author: Ido Efrat
author: Claudio Quadrelli
published in: Journal of Algebra
publication date: 2019-03-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1707.07018
review text: A major open problem in modern Galois theory is to characterize the profinite groups which are realizable as absolute Galois groups (briefly, AGG) of fields. It is well-known that if a profinite group has such a realization, then it is realizable as an AGG of a field of zero characteristic (see, e.g., Corollary~22.2.3 of the first-named author's book [\textit{I. Efrat}, Valuations, orderings, and Milnor \(K\)-theory. Providence, RI: AMS (2006; Zbl 1103.12002)]). Also, it follows from Galois cohomology that a profinite group \(G\) admits such a realization over a perfect field \(E_p\) of characteristic \(p\), for each prime number \(p\), if and only if \(G\) is a projective profinite group (see Chap. I, 5.9, Corollary~2, and Chap. II, 2.2, of [\textit{J.-P. Serre}, Galois cohomology. Berlin: Springer (1997; Zbl 0902.12004)] and Corollary~23.1.2 of [\textit{M. D. Fried} and \textit{M. Jarden}, Field arithmetic. 3rd rev. ed. Berlin: Springer (2008; Zbl 1145.12001)]). As shown by E. Artin (see Chap. VI, Corollary~9.3, of [\textit{S. Lang}, Algebra. 3rd rev. ed. New York, NY: Springer (2002; Zbl 0984.00001)]), AGG's do not contain elements of any finite order \(n \ge 3\). A more recent theorem due to Koenigsmann states that every solvable AGG is metabelian, which allows to classify profinite groups of this type, up-to (continuous) isomorphisms (see [\textit{J. Koenigsmann}, Invent. Math. 144, No. 1, 1--22 (2001; Zbl 1016.12005)]). In spite of the noted progress, and of other results concerning the stated major problem, it remains widely open. This attracts interest in the seemingly more accessible problem of characterizing, for a prime number \(p\), the pro-\(p\) groups \(G\) which are realizable as the maximal pro-\(p\) Galois group \(G_F (p)\) of some field \(F\) containing a primitive \(p\)-th root of unity. Among the known group-theoretic restrictions on pro-p groups \(G\) realizable as \(G_F (p)\) with \(F\) as above, one has Becker's pro-\(p\) version of the classi…
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 12E30
Mathematics Subject Classification ID: 12G05
zbMATH DE Number: 7034181
zbMATH Keywords: pro-\(p\) groups
zbMATH Keywords: Galois groups
zbMATH Keywords: Kummer theory
zbMATH Keywords: Galois cohomology
zbMATH Keywords: maximal pro-\(p\) Galois groups
zbMATH Keywords: cyclotomic pro-\(p\) pairs
MaRDI profile type: Publication
OpenAlex ID: W2736575451
arXiv ID: 1707.07018
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Wikidata QID: Q128452157
DOI: 10.1016/J.JALGEBRA.2019.01.015
description: scientific article