Galois groups of intersections of local fields

LetK be a denumerable Hilbertian field with separable algebraic closure and Galois group , letw ₁,...w _n be absolute values on . Then for almost allσ ∈ G _K ⁿ (in the sense of Haar measure) there are no relations between the decomposition groups G _K (ω ₁ σ ₁),...,G _K (w _n σ _n ) of the absolute valuesw ₁ σ ₁,...,w _n σ _n i.e. the subgroup of G _K generated by these groups is the free product of these groups.     

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.     

On absolute Galois splitting fields of central simple algebras

A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper proves an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. An analogous statement holds for division algebras over Henselian valued fields with global residue field.The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald-Wang theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings cannot be avoided by starting the construction with a symbol algebra.     

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).     

On Galois cohomology of semisimple groups over local and global fields of positive characteristic

We treat a case that was omitted from consideration in our article [2] in Math Zeit, 2007.     

On equivalent number fields with special Galois groups

In [12] and [13] Jack Sonn has introduced and studied a new notion of equivalence for number fields. In this note we show that “almost all” (cf. [14]) pairs of equivalent number fields are conjugate over ℚ, and we study equivalence classes of fields of prime degree.     

Solvable absolute Galois groups are metabelian

We show that solvable absolute Galois groups have an abelian normal subgroup such that the quotient is the direct product of two finite cyclic and a torsion-free procyclic group. In particular, solvable absolute Galois groups are metabelian. Moreover, any field with solvable absolute Galois group G admits a non-trivial henselian valuation, unless each Sylow-subgroup of G is either procyclic or isomorphic to Z ₂⋊Z/2Z. A complete classification of solvable absolute Galois groups (up to isomorphism) is given. Oblatum 22-IV-1998 & 1-IX-2000?Published online: 30 October 2000     

Quotients of absolute Galois groups which determine the entire Galois cohomology

For a prime power q = p ^d and a field F containing a root of unity of order q we show that the Galois cohomology ring H^*(G_F,\mathbbZ/q){H^*(G_F,\mathbb{Z}/q)} is determined by a quotient G_F^[3]{G_F^{[3]}} of the absolute Galois group G _F related to its descending q-central sequence. Conversely, we show that G_F^[3]{G_F^{[3]}} is determined by the lower cohomology of G _F . This is used to give new examples of pro-p groups which do not occur as absolute Galois groups of fields.     

On the number of certain Galois extensions of local fields

In this paper, we will calculate the number of Galois extensions of local fields with Galois group or .

     

On Galois cohomology of semisimple groups over local and global fields of positive characteristic, II

We show that the recent results of Prasad and Rapinchuk (Adv. Math. 207(2), 646–660, 2006) on the existence and uniqueness of certain global forms of semisimple algebraic groups with given local behaviour in the case of number fields still hold in the case of global function fields.     

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.     

On Galois cohomology of semisimple groups over local and global fields of positive characteristic, III

We extend some well-known results on Galois cohomology in its relation with weak approximation for connected linear algebraic groups over number fields to the case of global fields of positive characteristic. Some applications are considered.     

On Galois isomorphisms between ideals in extensions of local fields

LetL/K be a totally ramified, finite abelian extension of local fields, let and be the valuation rings, and letG be the Galois group. We consider the powers of the maximal ideal of as modules over the group ring . We show that, ifG has orderp ^m (withp the residue field characteristic), ifG is not cyclic (or ifG has orderp), and if a certain mild hypothesis on the ramification ofL/K holds, then and are isomorphic iffr≡r′ modp ^m . We also give a generalisation of this result to certain extensions not ofp-power degree, and show that, in the casep=2, the hypotheses thatG is abelian and not cyclic can be removed.     

Some Galois groups over number fields

Some conditions are stated which imply that certain finite groups are Galois groups over some number fields and related fields.     

Frobenius Galois groups over quadratic fields

There exists a quadratic fieldQ(√D) over which every Frobenius group is realizable as a Galois group.     

Galois groups over complete valued fields

We propose an elementary algebraic approach to the patching of Galois groups. We prove that every finite group is regularly realizable over the field of rational functions in one variable over a complete discrete valued field. Partially supported by NSF grant DMS 9306479.     

Computation of Galois groups over function fields

Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degree 8 as a Galois group over the rationals is included.