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.
2000 Mathematics Subject Classification: 12F12, 15A66.In this article we survey and examine the realizability of p-groups as Galois groups over arbitrary fields. In particular we consider various cohomological criteria that lead to necessary and sufficient conditions for the realizability of such a group as a Galois group, the embedding problem (i.e., realizability over a given subextension), descriptions of such extensions, automatic realizations among p-groups, and related topics.
The problem of the construction of number fields with Galois group over Q a given finite groups has made considerable progress in the recent years. The aim of this paper is to survey the current state of this problem, giving the most significant methods developed in connection with it.
Soit un corps et une extension quadratique de . Étant donné un polynôme de à groupe de Galois cyclique, nous donnons une méthode pour construire un polynôme de à groupe de Galois diédral, à partir des racines de . Cette méthode est tout à fait explicite : nous donnons de nombreux exemples de polynômes à groupe de Galois diédral sur le corps .
We give an effective characterization theorem for integral monic irreducible polynomials of degree whose Galois groups over are Frobenius groups with kernel of order and complement of prime order.
2000 Mathematics Subject Classification: 12F12We describe several types of Galois extensions having as Galois group the quaternion group Q16 of order 16.This work is partially supported by project of Shumen University.
In any normal number field having Q₈, the quaternion group of order 8, as Galois group over the rationals, at least two finite primes must ramify. The classical example by Dedekind of such a field is extraordinary in that it is totally real and only the primes 2 and 3 are ramified. In this note we describe in detail all Q₈-fields over the rationals where only two (finite) primes are ramified. We also show that, for any integer n>3 and any prime , there exist unique real and complex normal number...
On étudie différentes propriétés d’approximation pour des espaces homogènes (à stabilisateur fini) de sur un corps de nombres. On discute également du lien avec le problème de Galois inverse et on établit une formule pour le groupe de Brauer non ramifié de .
Soit un revêtement de la droite projective défini sur , de groupe de monodromie . Soit le compositum des corps de rationalité des points de branchement , et le corps des modules correspondants. Partant du lien entre corps des modules et espaces de Hurwitz, on étudie la géométrie et l’arithmétique de ces espaces et des espaces de configuration de points complétés pour évaluer la ramification dans des mauvaises places de qui ne divisent pas l’ordre de , mais où les points de branchements...
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.