Maximal Abelian Normal Subgroups of Galois Pro-2-Groups

Engler, Antonio José; Nogueira, Joao B.

doi:10.1006/jabr.1994.1164

Cited by 20 publications

(18 citation statements)

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“…Rigid elements have since been extensively used to detect valuations in large Galois groups. For instance, using rigid elements one can recover inertia/decomposition using the full relative pro-ℓ Galois theory of a field whose characteristic is prime to ℓ and which contains µ ℓ [EN94], [Efr95], [EK98]. Similar results also show how to recover inertia/decomposition in the absolute Galois group of an arbitrary field [Koe03].…”

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confidence: 96%

“…In a few words, the theory of rigid elements in the context of pro-ℓ Galois groups ( [EN94], [Efr95], [EK98]) asserts that the only way the situation above can arise is from valuation theory. More precisely, let K be a field such that char K = ℓ and µ ℓ ⊂ K. Suppose that σ, τ ∈ G K are non-torsion elements such that σ, τ = σ ⋊ τ is non-pro-cyclic.…”

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confidence: 99%

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Commuting-liftable subgroups of Galois groups II

Topaz

2015

Journal Für Die Reine Und Angewandte Mathematik

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Abstract. Let n denote either a positive integer or ∞, let ℓ be a fixed prime and let K be a field of characteristic different from ℓ. In the presence of sufficiently many roots of unity in K, we show how to recover some of the inertia/decomposition structure of valuations inside the maximal ℓ n -abelian Galois group of K using the maximal ℓ N -abelian-by-central Galois group of K, whenever N is sufficiently large relative to n.

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confidence: 96%

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confidence: 99%

Commuting-liftable subgroups of Galois groups II

Topaz

2015

Journal Für Die Reine Und Angewandte Mathematik

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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: 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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“…Remark 1.6. For other characterizations of decomposition groups -applicable to much more general situations -see, for example, Theorem 1.16 of [Pop 1994], Theorem 2 of [Koenigsmann 2003], or the results in [Engler and Koenigsmann 1998;Engler and Nogueira 1994].…”

Section: Generalities On Galois Groups Of Function Fields Of Curvesmentioning

confidence: 99%