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Bary-Soroker, Lior; Jarden, Moshe; Neftin, Danny

doi:10.1016/j.aim.2015.05.017

Cited by 11 publications

(6 citation statements)

References 22 publications

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“…On the other hand, since T ([y]) = (1, 1, a 1 ), we know from Lemma 4.1 that the generators σ 1 , σ 2 ∈ Gal(K/F ) extend to elements σ1 , σ2 ∈ Gal(K( √ y)/F ) that satisfy σ2 2 = σ2 1 = (σ 1 σ2 ) 4 = id. From this we see that Gal(K( √ y)/F ) Gal(K/F ) solves the embeddling problem D 4 Z/2Z⊕Z/2Z, where the kernel of the latter surjection is (σ 1 σ2 ) 2 .…”

Section: A Module Whose Fixed Part Complementsmentioning

confidence: 99%

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Galois module structure of square power classes for biquadratic extensions

Chemotti,

Minac,

Schultz

et al. 2021

Preprint

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This paper marks the first time that the Galois module structure of power classes of a field has been fully determined when the modular representation theory allows for an infinite number of indecomposable types. The example and methodology we consider pave the way for a paradigm shift in the study of absolute Galois groups via Galois modules. Whereas generic modules of this type are completely intractable to approach, this work shows that Galois cohomology provides a context where this otherwise impossible problem becomes approachable, interesting, and illuminating.Let V = Z/2Z⊕Z/2Z. For a Galois extension K/F with char(K) = 2 and Gal(K/F ) V , we determine the F 2 [Gal(K/F )]-module structure of K × /K ×2 . Although there are an infinite number of (pairwise non-isomorphic) indecomposable F 2 [V ]-modules, our decomposition includes at most 8 indecomposable types. This is a dramatic development on several fronts. Most conspicuously, it shows that these modules can be computed in settings which might initially seem too forbidding. The elegant nature of this decomposition also points to new structural restrictions on absolute Galois groups. More generally, the techniques used in this paper provide a blueprint for studying a broad class of Galois modules. This opens the door for a novel, systematic study of the Galois theory of maximal p-extensions of fields.Prepare for the unknown by studying how others in the past have coped with the unforeseeable and the unpredictable.

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Section: A Module Whose Fixed Part Complementsmentioning

confidence: 99%

“…(Note that it is also very interesting to study the p-Sylow subgroups of G K . The interested reader can consult [4].) The reader can also consult [77,78,79,80,81] to see systematic studies of certain explicit p-groups as Galois groups.…”

Section: Introductionmentioning

confidence: 99%

Galois module structure of square power classes for biquadratic extensions

Chemotti,

Minac,

Schultz

et al. 2021

Preprint

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“…[NSW08, Cor. 8.1.18]), it is an Iwasawa module, and the study of its structure is very important in algebraic number theory (see, e.g., [BSJN15]). From Theorem 6.6, one deduces that for every G/H-submodule V of H ab such that H ab /V is a finitely generated free abelian pro-p group, also the pair G L/K /V is Kummerian.…”

Section: Proposition 57 Let G = (G θ) Be a 1-smooth Torsion-free Cycl...mentioning

confidence: 99%

1-smooth pro-p groups and Bloch-Kato pro-p groups

Quadrelli

2019

Preprint

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Let p be a prime. We study pro-p groups G endowed with a continuous homomorphism G → 1+pZp satisfying a formal version of Hilbert 90. These pro-p groups are particularly important in Galois theory because by Kummer theory maximal pro-p Galois groups of fields containing a root of 1 of order p, together with the cyclotomic character, satisfy such property. In particular, we prove that De Clerq-Florence's "Smoothness Conjecture", which states that the Bloch-Kato conjecture follows from this formal version of Hilbert 90, holds true in the class of p-adic analytic pro-p groups.

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“…is also of importance (see e.g., [BSJN15]), however the corresponding cyclotomic pair G F /N p [N, N ] is not Kummerian. We thank the referee for pointing out these connections.…”

Section: Remarksmentioning

confidence: 99%

The Kummerian Property and Maximal Pro-$p$ Galois Groups

Efrat,

Quadrelli

2017

Preprint

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For a prime number p, we give a new restriction on pro-p groups G which are realizable as the maximal pro-p Galois group GF (p) for a field F containing a root of unity of order p. This restriction arises from Kummer Theory and the structure of the maximal p-radical extension of F . We study it in the abstract context of pro-p groups G with a continuous homomorphism θ : G → 1 + pZp, and characterize it cohomologically, and in terms of 1cocycles on G. This is used to produce new examples of pro-p groups which do not occur as maximal pro-p Galois groups of fields as above.

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The Sylow subgroups of the absolute Galois group Gal(Q)

Cited by 11 publications

References 22 publications

Galois module structure of square power classes for biquadratic extensions

Galois module structure of square power classes for biquadratic extensions

1-smooth pro-p groups and Bloch-Kato pro-p groups

The Kummerian Property and Maximal Pro-$p$ Galois Groups

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