Related papers: On the Definitions of Difference Galois Groups

Let $F$ be a $\delta-$field (differential field) of characteristic zero with an algebraically closed field of constants $F^\delta$, $A$ be a $\delta-F-$central simple algebra, $K$ be a Picard-Vessiot extension for the $\delta-F-$module $A$…

We develop the representation theory for reductive linear differential algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs, extending…

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…

We apply the difference-differential Galois theory developed by Hardouin and Singer to compute the differential-algebraic relations among the solutions to a second-order homogeneous linear difference equation of the form $…

We study torsors for groups defined by algebraic difference equations. Our main result provides necessary and sufficient conditions on the base difference field for all such torsors to be trivial. We also present an application to the…

We deal with aspects of the direct and inverse problems in parameterized Picard-Vessiot (PPV) theory. It is known that, for certain fields, a linear differential algebraic group (LDAG) G is a PPV Galois group over these fields if and only…

Let $F$ be a field with characteristic $\neq 2$. We show that $F$ is a nonrigid field if and only if certain small 2-groups occur as Galois groups over $F$. These results provide new "automatic realizability" results for Galois groups over…

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…

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…

In this article, we focus on orders in arbitrary number fields, consider their Picard groups and finally obtain ring class fields corresponding to them. The Galois group of the ring class field is isomorphic to the Picard group. As an…

The purpose of this short note is to establish the existence of $\partial$-parameterized Picard-Vessiot extensions of systems of linear difference-differential equations over difference-differential fields with algebraically closed…

This paper introduces a novel approach to understanding Galois theory, one of the foundational areas of algebra, through the lens of machine learning. By analyzing polynomial equations with machine learning techniques, we aim to streamline…

A self-contained exposition is given of the topological and Galois-theoretic properties of the category of combinatorial 1-complexes, or graphs, very much in the spirit of Stallings. A number of classical, as well as some new results about…

A finite group G is admissible over a field M if there is a division algebra whose center is M with a maximal subfield G-Galois over M. We consider nine possible notions of being admissible over M with respect to a subfield K of M, where…

We show that every linear algebraic group over an algebraically closed field of characteristic zero is the differential Galois group of a regular singular linear differential equation with rational function coefficients.

We consider (Frobenius) difference equations over (F_q(s,t), phi) where phi fixes t and acts on F_q(s) as the Frobenius endomorphism. We prove that every semisimple, simply-connected linear algebraic group G defined over F_q can be realized…

We study isomonodromicity of systems of parameterized linear differential equations and related conjugacy properties of linear differential algebraic groups by means of differential categories. We prove that isomonodromicity is equivalent…

We classify all cubic function fields over any finite field, particularly developing a complete Galois theory which includes those cases when the constant field is missing certain roots of unity. In doing so, we find criteria which allow…

Graded-division algebras are building blocks in the theory of finite-dimensional associative algebras graded by a group G. If G is abelian, they can be described, using a loop construction, in terms of central simple graded-division…

Topos properties of the category of covering groupoids over a fixed groupoid are discussed. A classification result for connected covering groupoids over a fixed groupoid analogous to the fundamental theorem of Galois theory is given.