Automatic realizability of Galois groups of order 16

This article examines the realizability of small groups of order , as Galois groups over arbitrary fields of characteristic not 2. In particular we consider automatic realizability of certain groups given the realizability of others.

     

Realizability and automatic realizability of Galois groups of order 32

This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.     

On Galois cohomology and realizability of 2-groups as Galois groups

In this paper we develop some new theoretical criteria for the realizability of p-groups as Galois groups over arbitrary fields. We provide necessary and sufficient conditions for the realizability of 14 of the 22 non-abelian 2-groups having a cyclic subgroup of index 4 that are not direct products of groups.     

On Galois cohomology and realizability of 2-groups as Galois groups II

In [Michailov I.M., On Galois cohomology and realizability of 2-groups as Galois groups, Cent. Eur. J. Math., 2011, 9(2), 403–419] we calculated the obstructions to the realizability as Galois groups of 14 non-abelian groups of order 2 ⁿ , n ≥ 4, having a cyclic subgroup of order 2 ⁿ⁻², over fields containing a primitive 2 ⁿ⁻³th root of unity. In the present paper we obtain necessary and sufficient conditions for the realizability of the remaining 8 groups that are not direct products of smaller groups.     

Galois realizability of groups of orders p 5 and p 6

Let p be an odd prime and k an arbitrary field of characteristic not p. We determine the obstructions for the realizability as Galois groups over k of all groups of orders p ⁵ and p ⁶ that have an abelian quotient obtained by factoring out central subgroups of order p or p ². These obstructions are decomposed as products of p-cyclic algebras, provided that k contains certain roots of unity.     

Approximating absolute Galois groups

In this paper we identify a class of profinite groups (totally torsion free groups) that includes all separable Galois groups of fields containing an algebraically closed subfield, and demonstrate that it can be realized as an inverse limit of torsion free virtually finitely generated abelian (tfvfga) profinite groups. We show by examples that the condition is quite restrictive. In particular, semidirect products of torsion free abelian groups are rarely totally torsion free. The result is of importance for K-theoretic applications, since descent problems for tfvfga groups are relatively manageable.     

Relatively projective groups as absolute Galois groups

By two well-known results, one of Ax, one of Lubotzky and van den Dries, a profinite group is projective iff it is isomorphic to the absolute Galois group of a pseudo-algebraically closed field. This paper gives an analogous characterization of relatively projective profinite groups as absolute Galois groups of regularly closed fields. Dedicated to Yuri Ershov on the occasion of his 60-th birthday Heisenberg-Stipendiat der Deutschen Forschungsgemeinschaft (KO 1962/1-1).     

Galois embeddings for linear groups

A criterion is given for the solvability of a central Galois embedding problem to go from a projective linear group covering to a vectorial linear group covering.

     

Small maximal pro-p Galois groups

For an odd primep we classify the pro-p groups of rank ≤4 which are realizable as the maximal pro-p Galois group of a field containing a primitive root of unity of orderp.     

Galois groups and complete domains

Consider a domain that is complete with respect to a non-zero prime ideal. This paper proves two Galois-theoretic results about such rings. Using Grothendieck’s Existence Theorem we prove that every finite group occurs as the Galois group of a Galois extension of . This generalizes results of David Harbater who proved the result in the case where the ideal is maximal and the domain is normal. As a consequence, we deduce that if is a Noetherian domain that is complete with respect to a non-zero prime ideal, then every finite group occurs as a Galois group over . This proves the Noetherian case of a conjecture posed by Moshe Jarden.     

Small maximal pro-p Galois groups

For an odd prime p we classify the pro-p groups of rank ≤ 4 which are realizable as the maximal pro-p Galois group of a field containing a primitive root of unity of order p. Received: 2 September 1997     

Galois groups via Atkin-Lehner twists

Using Serre's proposed complement to Shih's Theorem, we obtain as a Galois group over for at least new primes . Assuming that rational elliptic curves with odd analytic rank have positive rank, we obtain Galois realizations for of the primes that were not covered by previous results; it would also suffice to assume a certain (plausible, and perhaps tractable) conjecture concerning class numbers of quadratic fields. The key issue is to understand rational points on Atkin-Lehner twists of . In an appendix, we explore the existence of local points on these curves.

     

Motives with exceptional Galois groups and the inverse Galois problem

We construct motivic ?-adic representations of $\textup {Gal}(\overline {\mathbb{Q}}/\mathbb{Q})$ into exceptional groups of type E ₇,E ₈ and G ₂ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and is inspired by the Langlands correspondence for function fields. As an application, we solve new cases of the inverse Galois problem: the finite simple groups $E_{8}(\mathbb{F}_{\ell})$ are Galois groups over $\mathbb{Q}$ for large enough primes ?.     

Groups of p-absolute Galois type that are not absolute Galois groups

Let p be a prime. We study pro-p groups of p-absolute Galois type, as defined by Lam–Liu–Sharifi–Wake–Wang. We prove that the pro-p completion of the right-angled Artin group associated to a chordal simplicial graph is of p-absolute Galois type, and moreover it satisfies a strong version of the Massey vanishing property. Also, we prove that Demushkin groups are of p-absolute Galois type, and that the free pro-p product — and, under certain conditions, the direct product — of two pro-p groups of p-absolute Galois type satisfying the Massey vanishing property, is again a pro-p group of p-absolute Galois type satisfying the Massey vanishing property. Consequently, there is a plethora of pro-p groups of p-absolute Galois type satisfying the Massey vanishing property that do not occur as absolute Galois groups.