Detecting fast solvability of equations via small powerful Galois groups

Chebolu, Sunil K.; Mináč, Ján; Quadrelli, Claudio

doi:10.1090/s0002-9947-2015-06304-1

Cited by 18 publications

(15 citation statements)

References 38 publications

(33 reference statements)

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“…Besides fields F with G F (p) free pro-p-groups or Demushkin pro-p-groups, p-rigid fields play a fundamental role in current studies of Galois groups of maximal p-extensions. (See e.g., [6,13,47,48] and [49]. )…”

Section: Introductionmentioning

confidence: 99%

The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields

Mináč

Tân

2015

Advances in Mathematics

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A major difficult problem in Galois theory is the characterization of profinite groups which are realizable as absolute Galois groups of fields. Recently the Kernel n-Unipotent Conjecture and the Vanishing n-Massey Conjecture for n ≥ 3 were formulated. These conjectures evolved in the last forty years as a byproduct of the application of topological methods to Galois cohomology. We show that both of these conjectures are true for odd rigid fields. This is the first case of a significant family of fields where both of the conjectures are verified besides fields whose Galois groups of p-maximal extensions are free pro-p-groups. We also prove the Kernel Unipotent Conjecture for Demushkin groups of rank 2, and establish various filtration results for free pro-p-groups, provide examples of pro-p-groups which do not have the kernel n-unipotent property, compare various Zassenhaus filtrations with the descending p-central series and establish new type of automatic Galois realization.

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Section: Introductionmentioning

confidence: 99%

The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields

Mináč

Tân

2015

Advances in Mathematics

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“…Note that if G is a finitely generated pro-p group, then Φ 2 (G) is an open subgroup of G. Thus, the quotient G/Φ 2 (G) is finite, and one may reduce the equality Φ 2 (G) = P 3 (G) to a condition on finite p-groups, as done in [3,Corollary 4.15]. …”

Section: Therefore (3) Impliesmentioning

confidence: 99%

“…Also, let P n (G), n ≥ 1, denote the p-descending central series of G. In particular, one has P 2 (G) = Φ(G) and P 3 …”

Section: Introductionmentioning

confidence: 99%

“…Let F be a field containing a primitive p-th root of unity. By F × we denote the (multiplicative) group of non-zero elements of F. We consider the Galois extension F (3) of F obtained by first taking F (2) to be the compositum over F of all extensions of F of degree p, and then taking F (3) to be the compositum over F (2) of all the extensions of F (2) of degree p that are Galois over F. We also denote by F {3} the compositum over F (2) of all extensions of F (2) (2) ) (2) .…”

Section: Introductionmentioning

confidence: 99%

“…Sect. 2] and[3, Sect. 4.1]).Remark 1In the case p = 2, the Galois groups Gal(F (3) /F) and Gal(F {3} /F) are called W -group, resp.…”

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Finite quotients of Galois pro- $$p$$ p groups and rigid fields

Quadrelli

2015

Ann. Math. Québec

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For a prime number p, the author shows that if two certain canonical finite quotients of a finitely generated Bloch-Kato pro-p group G coincide, then G has a very simple structure, i.e., G is a p-adic analytic pro-p group (see Theorem 1). This result has a remarkable Galoistheoretic consequence: if the two corresponding canonical finite extensions of a field F-with F containing a primitive p-th root of unity-coincide, then F is p-rigid (see Corollary 1). The proof relies only on group-theoretic tools, and on certain properties of Bloch-Kato pro-p groups.Résumé Étant donné un nombre premier p, l'auteur montre que si G est un pro-p groupe de Bloch-Kato de type fini dont deux quotients finis canoniques coïncident, alors G admet une structure est très simple: c'est un pro-p groupe p-adique analytique (voir théorème 1). Ce résultat a une conséquence remarquable pour la théorie de Galois: si un corps F contient une racine primitive p-ième de l'unité et que ses deux extensions canoniques finies correspondantes coïncident, F est alors p-rigide (voir corollaire 1). La démonstration s'appuie sur des résultats de théorie des groupes et exploite certaines propriétés des pro-p groupes de Bloch-Kato.

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Oriented pro-$$\ell $$ groups with the Bogomolov–Positselski property

Quadrelli

Weigel

2022

Res. number theory

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For a prime number $$\ell $$ ℓ we say that an oriented pro-$$\ell $$ ℓ group $$(G,\theta )$$ ( G , θ ) has the Bogomolov–Positselski property if the kernel of the canonical projection on its maximal $$\theta $$ θ -abelian quotient $$\pi ^{\mathrm {ab}}_{G,\theta }:G\rightarrow G(\theta )$$ π G , θ ab : G → G ( θ ) is a free pro-$$\ell $$ ℓ group contained in the Frattini subgroup of G. We show that oriented pro-$$\ell $$ ℓ groups of elementary type have the Bogomolov–Positselski property (cf. Theorem 1.2). This shows that Efrat’s Elementary Type Conjecture implies a positive answer to Positselski’s version of Bogomolov’s Conjecture on maximal pro-$$\ell $$ ℓ Galois groups of a field $$\mathbb {K}$$ K in case that $$\mathbb {K}^\times /(\mathbb {K}^\times )^\ell $$ K × / ( K × ) ℓ is finite. Secondly, it is shown that for an $$H^\bullet $$ H ∙ -quadratic oriented pro-$$\ell $$ ℓ group $$(G,\theta )$$ ( G , θ ) the Bogomolov–Positselski property can be expressed by the injectivity of the transgression map $$d_2^{2,1}$$ d 2 2 , 1 in the Hochschild–Serre spectral sequence (cf. Theorem 1.4).

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Detecting fast solvability of equations via small powerful Galois groups

Abstract: 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 solvab… Show more

Cited by 18 publications

References 38 publications

The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields

The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields

Finite quotients of Galois pro- $$p$$ p groups and rigid fields

Oriented pro-$$\ell $$ groups with the Bogomolov–Positselski property

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