This paper surveys recent work on the problems of calculating Galois groups of diierentialequations and of constructingdiierentialequations with given groups as their Galois groups.

We investigate the properties of Galois, dual Galois, residuated, and dual residuated connections on posets. In particular, we show that their connections are related to relations.

This paper considers some refined versions of the Inverse Galois Problem. We study the local or global behavior of rational specializations of some finite Galois covers of PQ.

We compute a polynomial with Galois group SL 2 (11) over Q.Furthermorewe prove that SL 2 (11) is the Galois group of a regular extension of Q(t).

Recall that the birational anabelian conjecture originating in ideas presented in Grothendieck’s Esquisse d’un Programme [G1] and Letter to Faltings [G2], asserts roughly the following: First, there should exist a group theoretical recipe by which one can recognize the absolute Galois groups GK of finitely generated infinite fields K among all the profinite groups. Second, if G = GK is such an ...

l-adic Étale Theory Hodge Theory Category of l-adic Abelian category MHR Galois modules of real mixed Hodge strucrures Galois group Hodge Galois group GHod := Gal(Q/Q) Galois group of the category MHR Gal(Q/Q) acts on H∗ et(X,Ql), H∗(X(C),R) has a functorial where X is a variety over Q real mixed Hodge structure étale site ?? Gal(Q/Q) acts on the étale site, and thus ?? on categories of étale s...

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 gro...

This article is made up with two parts. In the first part, using a recent result of Schauenburg, one generalizes to the case when objects are faithfully flat over the ground ring, the full equivalence between the notions of Hopf-Galois objects and Hopf-Galois systems. In this last description, one gives explicitly an inverse for a Hopf-Galois object T together with its generalized antipode. In ...

In this article new cases of the inverse Galois problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL2(Fpn ) occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are cons...

Quantum systems in which the position and momentum take values in the ring Zd and which are described with d-dimensional Hilbert space, are considered. When d is the power of a prime, the position and momentum take values in the Galois field GF (p), the position-momentum phase space is a finite geometry and the corresponding ‘Galois quantum systems’ have stronger properties. The study of these ...