Related papers: A formulation of difference Galois theory
A conceptual framework for variational formulations of physical theories is proposed. Such a framework is displayed here just for statics, but it is designed to be subsequently adapted to variational formulations of static field theories…
In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which…
We count the number of Galois extensions $M/\mathbb{Q}$ with fixed Galois group $\text{Gal}(M/\mathbb{Q})=D_4$ ordered by multi-invariants introduced by Gundlach. We verify the asymptotic behavior predicted by Gundlach's version of Malle's…
Differential equations have arithmetic analogues in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations and the present paper is concerned with the "linear" ones. The equations…
This paper develops from scratch a theory of Galois rings and orders over arbitrary fields. Our approach is different from others in the literature in that there is no non-modularity assumption. We prove, when the field is algebraically…
We study the relationship between the local and global Galois theory of function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and…
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…
This is a guide to the construction of nonlinear number fields, which includes new points not found in our earlier article ``Geometric Galois theory, nonlinear number fields and a Galois group interpretation of the idele class group''.
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…
In this note we give a counter example to a conjecture of Malle which predicts the asymptotic behaviour of the counting functions for field extensions with given Galois group and bounded discriminant.
The aim of the paper is to introduce B-extensions which are the most symmetrical finite field extensions (a finite field extension $L/K$ is called a {\it B-extension} if the endomorphism algebra ${\rm End}_K(L)$ is generated by the algebra…
In this paper, we present methods to simplify reducible linear differential systems before solving. Classical integrals appear naturally as solutions of such systems. We will illustrate the methods developed in a previous paper on several…
It is quite natural to wonder whether there is a difference-differential equations, the Galois group of which is a quantum group that is neither commutative nor co-commutative. Believing that there was no such linear equations, we explored…
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…
In this paper we consider the problem of computing the difference Galois groups of order three equations for a large class of difference operators including the shift operator (Case S), the $q$-difference operator (Case Q), the Mahler…
We develop a new connection between Differential Algebra and Geometric Invariant Theory, based on an anti-equivalence of categories between solution algebras associated to a linear differential equation (i.e. differential algebras generated…
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…
Intended for mathematical physicists interested in applications of the division algebras to physics, this article highlights some of their more elegant properties with connections to the theories of Galois fields and quadratic residues.
In this talk we describe our recent work on discrete quantum theory based on Galois fields. In particular, we discuss how discrete quantum theory sheds new light on the foundations of quantum theory and we review an explicit model of…
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.