Related papers: On differential Galois groups of strongly normal e…

One of the key points in Galois theory via field extensions is to build up a correspondence between subfields of a field and subgroups of its automorphism group, so as to study fields via methods of groups. As an analogue of the Galois…

As a simple corollary of a highly general framework for differential and difference Galois theory introduced by Y. Andre, we formulate a version of the Galois correspondence that applies over a difference field with arbitrary field of…

Jacobson developed a counterpart of Galois theory for purely inseparable field extensions in positive characteristic. In his theory, a certain type of derivations replace the role of the generators of Galois groups. This article provides a…

Let $G$ be a classical group of dimension $d$ and let $\boldsymbol{a}=(a_1,\dots,a_d)$ be differential indeterminates over a differential field $F$ of characteristic zero with algebraically closed field of constants $C$. Further let…

A differential version of the classical Weil descent is established in all characteristics. It yields a theory of differential restriction of scalars for differential varieties over finite differential field extensions. This theory is then…

This paper is a finishing touch to the (over 200 years) {\em classical} `Galois Theory' of {\em arbitrary} finite field extensions, i.e. the goal of it is to describe intermediate subfields of an arbitrary finite field extension via {\em…

We give a description of the rational representations of the differential Galois group of a Picard-Vessiot extension.

We introduce a notion of "Galois closure" for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an S_n degree n extension of fields. Moreover, we prove a number of properties of this…

This article is interested in pullbacks under the logarithmic derivative of algebraic ordinary differential equations. In particular, assuming the solution set of an equation is internal to the constants, we would like to determine when its…

In this paper we give a unified approach in categorical setting to the problem of finding the Galois closure of a finite cover, which includes as special cases the familiar finite separable field extensions, finite unramified covers of a…

We develop a Galois theory for difference ring extensions, inspired by Magid's separable Galois theory for ring extensions and by Janelidze's categorical Galois theory. Our difference Galois theorem states that the category of difference…

Let F be a differential field of characteristic zero. In this article, we construct Picard-Vessiot extensions of F whose differential Galois group is isomorphic to the full unipotent subgroup of the upper triangular group defined over the…

This PhD deals with the notion of pseudo algebraically closed (PAC) extensions of fields. It develops a group-theoretic machinery, based on a generalization of embedding problems, to study these extensions. Perhaps the main result is that…

The purpose of this paper is to constructively develop a Galois theory on irreducible shifts of finite type (SFTs) and to analyze the automorphism groups of SFTs using this framework. Let $X$ and $Y$ be irreducible SFTs. We demonstrate that…

In this paper we produce unconditionally new instances of Galois number field extensions exhibiting strong discrepancies in the distribution of Frobenius elements among conjugacy classes of the Galois group. We first prove an inverse Galois…

In 2018, Legrand and Paran proved a weaker form of the Inverse Galois Problem for all Hilbertian fields and all finite groups: that is, there exist possibly non-Galois extensions over given Hilbertian base field with given finite group as…

In this paper, we develop a difference Galois theory in the setting of real fields. After proving the existence and uniqueness of the real Picard-Vessiot extension, we define the real difference Galois group and prove a Galois…

These are the notes for an undergraduate course at the University of Edinburgh, 2021-2023. Assuming basic knowledge of ring theory, group theory and linear algebra, the notes lay out the theory of field extensions and their Galois groups,…

We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as the Galois group of a Galois extension of $\mathbb{Q}$ all of whose decomposition groups are cyclic (resp., abelian). This…

A categorical theory for the discretization of a large class of dynamical systems with variable coefficients is proposed. It is based on the existence of covariant functors between the Rota category of Galois differential algebras and…