On the characterization of local fields by their absolute Galois groups

For finite field extensions of the field of Henselian p-adic rational numbers necessary and sufficient conditions are given which state that the fields have isomorphic absolute Galois groups; it is thereby supposed that a p-th root of unity (a 4-th when p = 2) belongs to the fields. Also examples are discussed.     

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.

     

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.     

Densities of quartic fields with even Galois groups

Let be the number of degree number fields with Galois group and whose discriminant satisfies . Under standard conjectures in diophantine geometry, we show that , and that there are monic, quartic polynomials with integral coefficients of height whose Galois groups are smaller than , confirming a question of Gallagher. Unconditionally we have , and that the -class groups of almost all Abelian cubic fields have size . The proofs depend on counting integral points on elliptic fibrations.

     

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.     

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.     

Actions of Galois groups on invariants of number fields

In this paper we investigate the connection between relations among various invariants of number fields LH corresponding to subgroups H acting on L and of linear relations among norm idempotents.     

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.     

Minimal discriminants for fields with small Frobenius groups as Galois groups

We apply class field theory to the computation of the minimal discriminants for certain solvable groups. In particular, we apply our techniques to small Frobenius groups and all imprimitive degree 8 groups such that the corresponding fields have only a degree 2 and no degree 4 subfield.     

Galois cohomology of the classical groups over imperfect fields

Elaborating on techniques of Bayer-Fluckiger and Parimala, we prove the following strong version of Serre’s Conjecture II for classical groups: let G be a simply connected absolutely simple group of outer type A_n or of type B_n, C_n or D_n (non trialitarian) defined over an arbitrary field F. If the separable dimension of F is at most 2 for every torsion prime of G, then every G-torsor is trivial.     

Isomorphisms of Galois groups of number fields with restricted ramification

Let K be a number field and S a set of primes of K. We write $K_S/K$ for the maximal extension of K unramified outside S and $G_{K,S}$ for its Galois group. In this paper, we answer the following question under some assumptions: “For $i=1,2$, let $K_i$ be a number field, $S_i$ a (sufficiently large) set of primes of $K_i$ and $\sigma :G_{K_1,S_1}\stackrel{\sim }{\to }{G}_{K_2,S_2}$ an isomorphism. Is σ induced by a unique isomorphism between $K_{1,S_1}/K_1$ and $K_{2,S_2}/K_2$?” Here, the main assumption is about the Dirichlet density of $S_i$.     

Abelian-by-central Galois groups of fields II: Definability of inertia/decomposition groups

This paper explores some first-order properties of commuting-liftable pairs in pro-? abelian-by-central Galois groups of fields. The main focus of the paper is to prove that minimized inertia and decomposition groups of many valuations are first-order definable using a predicate for the collection of commuting-liftable pairs. For higher-dimensional function fields over algebraically closed fields, we show that the minimized inertia and decomposition groups of quasi-divisorial valuations are uniformly first-order definable in this language.     

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 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.