Related papers: Splitting fields and general differential Galois t…

We introduce the existence of a Genus-Type Theory that generalizes classical genus theory by linking fractional ideals of number fields to structures built from their Galois groups and associated Diophantine equations, as formally stated in…

This paper introduces a natural extension of Kolchin's differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard-Vessiot theory recently developed by Matzat…

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…

Let $G$ be an algebraic group, $X$ a generically free $G$-variety, and $K=k(X)^G$. A field extension $L$ of $K$ is called a splitting field of $X$ if the image of the class of $X$ under the natural map $H^1(K, G) \mapsto H^1(L, G)$ is…

The author surveys Galois theory of function fields with non-zero caracteristic and its relation to the structure of finite permutation groups and matrix groups.

The outlines of a "Galois theory" for bimeromorphic geometry is here developed, via the study of model-theoretic definable binding groups in the theory CCM of compact complex spaces. As an application, a structure theorem about principal…

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 present a geometric setting for the differential Galois theory of $G$-invariant connections with parameters. As an application of some classical results on differential algebraic groups and Lie algebra bundles, we see that the Galois…

We develop a Galois theory for systems of linear difference equations with an action of an endomorphism {\sigma}. This provides a technique to test whether solutions of such systems satisfy {\sigma}-polynomial equations and, if yes, then…

Generalized non-autonomous linear celullar automata are systems of linear difference equations with many variables that can be seen as convolution equations in a discrete group. We study those systems from the stand point of the Galois…

We give a computationally efficient method for constructing the linear differential operator with polynomial coefficients whose space of holomorphic solutions is spanned by all the branches of a function defined by a generic algebraic…

We enhance the analogy between field extensions and covering spaces by introducing the concept of splitting covering which correspondences to the splitting field in Galois theory. We define semi-topological Galois groups for Weierstrass…

There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of…

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of {\em Colombeau type} in the sense that it contains a copy of the space of Schwartz…

Hopf Galois theory expands the classical Galois theory by considering the Galois property in terms of the action of the group algebra k[G] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable…

We study a necessary condition for the integrability of the polynomials fields in the plane by means of the differential Galois theory. More concretely, by means of the variational equations around a particular solution it is obtained a…

We construct a Galois correspondence for finite purely inseparable field extensions $F/K$, generalising a classical result of Jacobson for extensions of exponent one (where $x^p \in K$ for all $x\in F$).

In this article, we realize some groups as Galois groups over rational numbers and finite extension of rational numbers by studying right splitting of some exact sequences, Galois correspondence and algebraic operations on Galois…

This note presents Galois theory for finite fields. It was written as a handout for the MAT401 course ``Polynomial equations and fields'' taught at the University of Toronto in Spring 2026. We use without proofs some basic properties of…

In this paper, we explain how to compute the Lie algebra of the differential Galois group of a reducible linear differential system. We achieve this by showing how to transform a block-triangular linear differential system into a…