The Kummerian property and maximal pro-\(p\) Galois groups
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 classical Artin-Schreier theorem: finite subgroups of \(G\) can only be trivial or of order \(2\) (see [\textit{E. Becker}, J. Reine Angew. Math. 268/269, 41--52 (1974; Zbl 0289.12103)]). Also, in case \(G \neq \{1\}\), the additive group \(\mathbb{Z}_{p}\) of \(p\)-adic integers is a homomorphic image of \(G\) unless \(F\) is a Pythagorean field, i.e. \(F\) is formally real, in the sense of Artin and Schreier, and the set of squares of elements of \(F\) is closed under addition; moreover, if \(p > 2\), then \(G\) is a torsion-free group (see [\textit{G. Whaples}, Duke Math. J. 24, 201--204 (1957; Zbl 0081.26703)]). Next, it follows from the deep results of Rost and Voevodsky that the cohomology ring \(H^{\ast}(G, Z/p)\) is quadratic, i.e., it is generated by degree \(1\) elements and its relations originate from the degree \(2\) part (see Remark~8.2 of [\textit{S. K. Chebolu} et al., Math. Ann. 352, No. 1, 205--221 (2012; Zbl 1272.12015)] and Sect. 2 of [\textit{C. Quadrelli}, Forum Math. 26, No. 3, 793--814 (2014; Zbl 1308.20029)]). The paper under review gives a new restriction on pro-\(p\) groups \(G\) which are realizable as the maximal pro-p Galois group \(G_F (p)\), for a field \(F\) as above. This restriction arises from Kummer Theory and the structure of the maximal \(p\)-radical extension of \(F\). The authors study it in the abstract context of pro-\(p\) groups \(G\) with a continuous homomorphism \(\theta : G \to 1 + p\mathbb{Z}_{p}\), and characterize it cohomologically, and in terms of \(1\)-cocycles on \(G\). This is used for constructing new examples of pro-\(p\) groups which do not occur as maximal pro-\(p\) Galois groups of the considered fields. As shown by the authors, the new restriction implies the Artin-Schreier/Becker restriction on the finite subgroups of \(G_F (p)\), but is independent of the cohomological quadraticness restriction, as well as of the vanishing property of \(3\)-fold Massey products (see [\textit{E. Matzri}, ``Triple Massey products in Galois cohomology'', Preprint, \url{arXiv:1411.4146}], [\textit{J. Mináč} and \textit{N. D. Tân}, J. Lond. Math. Soc., II. Ser. 94, No. 3, 909--932 (2016; Zbl 1378.12002)] and [\textit{I. Efrat} and \textit{E. Matzri}, J. Eur. Math. Soc. 19, No. 12, 3629--3640 (2017; Zbl 1425.12004)]).
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