Abelian subgroups of pro-$p$ Galois groups

Engler, Antonio José; Koenigsmann, Jochen

doi:10.1090/s0002-9947-98-02063-7

Cited by 33 publications

(26 citation statements)

References 15 publications

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“…As a corollary of Theorem 1.4, we obtain yet another well-known result on maximal pro-p Galois groups due to Engler and Nogueira [15] (for p = 2) and Engler and Koenigsmann [14] (for p > 2).…”

Section: Our First Substantial Results Issupporting

confidence: 58%

Frattini-injectivity and Maximal pro-$p$ Galois groups

Snopce¹,

Tanushevski²

2020

Preprint

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We call a pro-p group G Frattini-injective if distinct finitely generated subgroups of G have distinct Frattinis. This paper is an initial effort toward a systematic study of Frattini-injective pro-p groups (and several other related concepts). Most notably, we classify the p-adic analytic and the solvable Frattini-injective pro-p groups, and we describe the lattice of normal abelian subgroups of a Frattini-injective pro-p group.We prove that every maximal pro-p Galois group of a field that contains a primitive pth root of unity (and also contains √ −1 if p = 2) is Frattiniinjective. In addition, we show that many substantial results on maximal pro-p Galois groups are in fact consequences of Frattini-injectivity. For instance, a p-adic analytic or solvable pro-p group is Frattini-injective if and only if it can be realized as a maximal pro-p Galois group of a field that contains a primitive pth root of unity (and also contains √ −1 if p = 2); and every Frattini-injective pro-p group contains a unique maximal abelian normal subgroup.

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Section: Our First Substantial Results Issupporting

confidence: 58%

Frattini-injectivity and Maximal pro-$p$ Galois groups

Snopce¹,

Tanushevski²

2020

Preprint

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“…(An element of the form n a + m √ b, with a, b ∈ F {0} and n, m > 1 is, for instance, nested.) This improves the following result: the Galois group of the maximal p-extension F (p)/F is a solvable group if and only if the field F is p-rigid (proved first in [EK98]).…”

Section: Introductionsupporting

confidence: 54%

Detecting fast solvability of equations via small powerful Galois groups

Chebolu

Mináč

Quadrelli

2015

Trans. Amer. Math. Soc.

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Fix an odd prime p p , and let F F be a field containing a primitive p p th root of unity. It is known that a p p -rigid field F F is characterized by the property that the Galois group G F ( p ) G_F(p) of the maximal p p -extension F ( p ) / F F(p)/F is a solvable group. We give a new characterization of p p -rigidity which says that a field F F is p p -rigid precisely when two fundamental canonical quotients of the absolute Galois groups coincide. This condition is further related to analytic p p -adic groups and to some Galois modules. When F F is p p -rigid, we also show that it is possible to solve for the roots of any irreducible polynomials in F [ X ] F[X] whose splitting field over F F has a p p -power degree via non-nested radicals. We provide new direct proofs for hereditary p p -rigidity, together with some characterizations for G F ( p ) G_F(p) – including a complete description for such a group and for the action of it on F ( p ) F(p) – in the case F F is p p -rigid.

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“…In Galois theory one has the following result, due to A. Engler, J. Koenigsmann and J. Nogueira (cf. [10] and [11]). Theorem 3.2 Let K be a field containing a root of 1 of order p (and also √ −1 if p = 2), and suppose that the maximal pro-p Galois group G K (p) of K is not isomorphic to Z p .…”

Section: 2mentioning

confidence: 98%

“…This subgroup is abelian, and normal in G. In [11], A. Engler and J. Koenigsmann showed that if the maximal pro-p Galois group G K (p) of a field K is not cyclic then it has a unique maximal normal abelian closed subgroup (i.e., one containing all normal abelian closed subgroups of G K (p)), which coincides with the θ K -center Z(G K ), and the short exact sequence of pro-p groups…”

Section: Quadrellimentioning

confidence: 99%

Galois-theoretic features for 1-smooth pro-p groups

Quadrelli

2021

Can. Math. Bull.

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Let p be a prime. A pro-p group G is said to be 1-smooth if it can be endowed with a continuous representation θ : G → GL 1 (Z p ) such that every open subgroup H of G, together with the restriction θ | H , satisfies a formal version of Hilbert 90. We prove that every 1-smooth pro-p group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-p Galois groups of fields, and that if a 1-smooth pro-p group is solvable, then it is locally uniformly powerful, in analogy with a result by Ware on maximal pro-p Galois groups of fields. Finally we ask whether 1-smooth pro-p groups satisfy a "Tits' alternative".

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Abelian subgroups of pro-$p$ Galois groups

Cited by 33 publications

References 15 publications

Frattini-injectivity and Maximal pro-$p$ Galois groups

Frattini-injectivity and Maximal pro-$p$ Galois groups

Detecting fast solvability of equations via small powerful Galois groups

Galois-theoretic features for 1-smooth pro-p groups

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