This work presents a difference geometric approach to the strongly normal Galois theory of difference equations. In this approach, a system of ordinary difference equations is encoded in a difference extension, and the Galois groups are group schemes of finite type over the constants. The Galois groups need neither be linear nor reduced. The main result is a characterization of the extensions t...

Picard-Vessiot rings are present in many settings like differential Galois theory, difference Galois theory and Galois theory of Artinian simple module algebras. In this article we set up an abstract framework in which we can prove theorems on existence and uniqueness of Picard-Vessiot rings, as well as on Galois groups corresponding to the Picard-Vessiot rings. As the present approach restrict...

— The paper consists of two parts. In the first part, we explain an excellent idea, due to mathematicians of the 19-th century, of naturally developing classical Galois theory of algebraic equations to an infinite dimensional Galois theory of nonlinear differential equations. We show with an instructive example how we can realize the idea of the 19-th century in a rigorous framework. In the sec...

Differential Galois Theory is a branch of abstract algebra that studies fields equipped with a derivation function. In much the same way as ordinary Galois Theory studies field extensions generated by solutions of polynomials over a base field, differential Galois Theory studies differential field extensions generated by solutions to differential equations over a base field. In this paper, we w...

We establish automatic realizations of Galois groups among groups M ⋊ G, where G is a cyclic group of order p for a prime p and M is a quotient of the group ring Fp[G]. The fundamental problem in inverse Galois theory is to determine, for a given field F and a given profinite group G, whether there exists a Galois extension K/F such that Gal(K/F ) is isomorphic to G. A natural sort of reduction...

10. Galois modules and class field theory Boas Erez In this section we shall try to present the reader with a sample of several significant instances where, on the way to proving results in Galois module theory, one is lead to use class field theory. Conversely, some contributions of Galois module theory to class fields theory are hinted at. We shall also single out some problems that in our op...

Partial Galois extensions were recently introduced by Doku-chaev, Ferrero and Paques. We introduce partial Galois extensions for noncommutative rings, using the theory of Galois corings. We associate a Morita context to a partial action on a ring.

Let N be a bijective measure space. D. Sasaki’s classification of positive definite, globally universal domains was a milestone in homological Galois theory. We show that |O| ≤ K. The groundbreaking work of H. Thomas on contra-arithmetic planes was a major advance. Recent developments in analytic group theory [36, 36] have raised the question of whether Fréchet’s conjecture is false in the cont...

Differential Galois theory has known an outburst of activity in the last decade. To pinpoint what triggered this renewal is probably a matter of personal taste; all the same, let me start the present review by a tentative list, restricted on purpose to “non-obviously differential” occurrences of the theory (and also, as in the book under review, to the Galois theory of linear differential equat...

D ifferential Galois theory, like the more familiar Galois theory of polynomial equations on which it is modeled, aims to understand solving differential equations by exploiting the symmetry group of the field generated by a complete set of solutions to a given equation. The subject was invented in the late nineteenth century, and by the middle of the twentieth had been recast in modern rigorou...