Polynomials with PSL(2, 7) as Galois group

We describe a technique for determining the set-transitivity of the Galois group of a polynomial over the rationals. As an application we give a short proof that the polynomial P₇(x) = x⁷ ? 154x + 99 has the simple group PSL(2, 7) of order 168 as its Galois group over the rationals. A similar method is used to prove that the associated splitting field is not that of the polynomial x⁷ ? 7x + 3 given by Trinks [9].     

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.

     

Galois theory, discriminants and torsion subgroup of elliptic curves

We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the discriminant of the elliptic curve.     

Weyl groups as Galois groups of a regular extension of the field ℚ

For a finite group G, Gal_T(G) denotes the property that there exists a regular Galois extension of the rational function field ℚ(T) over the field of rationals ℚ, with a Galois group G. This property is established to be satisfied by all Weyl groups except the type F₄, from which it follows that it holds also for Chevalley groups C₃(2) and D₄(2). Translated fromAlgebra i Logika, Vol. 34, No. 3, pp. 311-315, May-June, 1995.     

Indépendance Linéaire sur Q de logarithmes P-adiques de nombres algébriques et rang P-adique du groupe des unités d'un corps de nombres

Following Ax's method a lower bound for the p-adic rank of the group of units in the general case of a Galois number field over the rationals is given. In some nontrivial special cases the result gives Leopoldt's conjecture. The same method is also applied to the case of p-units.     

Integer Polynomials with Roots mod p for all Primes p

Let f(X) be an integer polynomial which is a product of two irreducible factors. Assume that f(X) has a root mod p for all primes p. If the splitting field of f(X) over the rationals is a cyclic extension of the stem fields, then the Galois group of f(X) over the rationals is soluble and of bounded Fitting length. Moreover, the fixed groups of the stem extensions are in, some sense, unique.     

Zur Berechnung der Kapitulation in bikubischen Zahlkörpern

The capitulation kernel is the kernel of the natural extension homomorphism of the ideal class groups in a extension K|k of number fields. In this paper K is a non-cyclic Galois field of degree 6 over the rationals and k is its quadratic subfield. Two different methods of computing the capitulation kernel are discussed. Both depend on the relationship between capitulation and unit structure. The paper closes with two tables. They contain the capitulation kernel for all ramified extensions K|k having cubic discriminants between –20000 and 100000.     

Irreducible characters of 3′-degree of finite symmetric,general linear and unitary groups

Let G be a finite symmetric, general linear, or general unitary group defined over a field of characteristic coprime to 3. We construct a canonical correspondence between irreducible characters of degree coprime to 3 of G and those of ${\mathbf{N}}_G(P)$, where P is a Sylow 3-subgroup of G. Since our bijections commute with the action of the absolute Galois group over the rationals, we conclude that fields of values of character correspondents are the same.     

The algebraic degree of geometric optimization problems

In this paper we apply Galois methods to certain fundamentalgeometric optimization problems whose exact computational complexity has been an open problem for a long time. In particular we show that the classic Weber problem, along with theline-restricted Weber problem and itsthree-dimensional version are in general not solvable by radicals over the field of rationals. One direct consequence of these results is that for these geometric optimization problems there existsno exact algorithm under models of computation where the root of an algebraic equation is obtained using arithmetic operations and the extraction ofkth roots. This leaves only numerical or symbolic approximations to the solutions, where the complexity of the approximations is shown to be primarily a function of the algebraic degree of the optimum solution point.     

On the validity of a Benz condition for non-standard Galois fields and for 2e

W. Benz showed that if a field K satisfies certain conditions, then every injection K² K² preserving Minkowsky distance 1 is semilinear up to translation. In this paper we consider one of these conditions. We prove that a non-standard Galois field associated with a Cauchy ultra filter over the set of all prime powers satisfies Benz condition but that the field of rationals does not. The last statement is proved by studying the set of rational points of a certain quartic curve, by considering a Weierstrass cubic birationally equivalent to that quartic.Dedicated to Professor M. Scafati Tallini on the occasion of her 65^th birthday     

The Intrinsic Fundamental Group of a Linear Category

We provide an intrinsic definition of the fundamental group of a linear category over a ring as the automorphism group of the fibre functor on Galois coverings. If the universal covering exists, we prove that this group is isomorphic to the Galois group of the universal covering. The grading deduced from a Galois covering enables us to describe the canonical monomorphism from its automorphism group to the first Hochschild-Mitchell cohomology vector space.     

Arithmetic of quasi-cyclotomic fields

We call a quadratic extension of a cyclotomic field a quasi-cyclotomic field if it is non-abelian Galois over the rational number field. In this paper, we study the arithmetic of any quasi-cyclotomic field, including to determine the ring of integers of it, the decomposition nature of prime numbers in it, and the structure of the Galois group of it over the rational number field. We also describe explicitly all real quasi-cyclotomic fields, namely, the maximal real subfields of quasi-cyclotomic fields which are Galois over the rational number field. It gives a series of totally real fields and CM fields which are non-abelian Galois over the rational number field.     

On group orders of rational points of elliptic curves

We consider elliptic curves without complex multiplication defined over the rationals or with complex multiplication defined over the Hilbert class field of the endomorphism ring. We examine the distribution of almost prime group orders of these curves when reduced modulo a prime ideal.     

On σδ-Picard–Vessiot Extensions

We study the differential Galois theory of difference equations under weaker hypothesis on the field of σ-constants. This framework yields a new approach to results by C. Hardouin and M. Singer, which answers positively a question by M. Singer: under the classical hypothesis, the known results are still valid. In particular, our Galois group is isomorphic to theirs over a suitable field. We also explicitly calculate the number of connected components of the Galois group.     

Normal integral bases for extensions of the rationals

We give an algorithm for constructing normal integral bases of tame Galois extensions of the rationals with group . Using earlier works we can do the same until degree .

     

On the number of integral ideals in Galois extensions

Ifa _k denotes the number of integral ideals with normk, in any finite Galois extension of the rationals, we study sums of the form \(\sum\limits_{k \leqslant x} {a_k^l } (l = 2,3, \ldots )\) , along with the integral means of the 2?-th power (? real, ?≥1) of the absolute value of the corresponding Dedekind zeta-function. The two averages are related if ?=n ^1?1/2, wheren is the degree of the Galois extension.     

Morita contexts and Galois theory for weak Hopf comodulelike algebras

We investigate the Morita context and graded cases for weak group corings and derive some equivalent conditions for μ to be surjective. Furthermore, we develop Galois theory for weak group corings. As an application, we give Galois theory for comodulelike algebras over a weak Hopf group coalgebra.     

Infinitesimal group schemes as iterative differential Galois groups

This article is concerned with Galois theory for iterative differential fields (ID-fields) in positive characteristic. More precisely, we consider purely inseparable Picard-Vessiot extensions, because these are the ones having an infinitesimal group scheme as iterative differential Galois group. In this article we prove a necessary and sufficient condition to decide whether an infinitesimal group scheme occurs as Galois group scheme of a Picard-Vessiot extension over a given ID-field or not. In particular, this solves the inverse ID-Galois problem for infinitesimal group schemes. Furthermore, this gives a tool to tell whether all purely inseparable ID-extensions are in fact Picard-Vessiot extensions.     

Group Corings

We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to graded modules over graded rings. Galois group corings are defined, and a Structure Theorem for the G-comodules over a Galois group coring is given. We study (graded) Morita contexts associated to a group coring. Our theory is applied to group corings associated to a comodule algebra over a Hopf group coalgebra. This research was supported by the research project G.0622.06 “Deformation quantization methods for algebras and categories with applications to quantum mechanics” from Fonds Wetenschappelijk Onderzoek-Vlaanderen. The third author was partially supported by the SRF (20060286006) and the FNS (10571026).