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We show how the Galois-Picard_Vessiot theory of differential equations and difference equations, and the theory of holonomy groups in differential geometry, are different aspects of a unique Galois theory. The latter is based upon the…

We show that a finite-dimensional tame division algebra D over a Henselian field F has a maximal subfield Galois over F if and only if its residue division algebra has a maximal subfield Galois over the residue field of F. This generalizes…

This paper lays the foundations for a nonlinear theory of differential geometry that is developed in a subsequent paper which is based on Colombeau algebras of tensor distributions on manifolds. We adopt a new approach and construct a…

In this exposition we discuss the theory of algebraic extensions of valued fields. Our approach is mostly through Galois theory. Most of the results are well-known, but some are new. No previous knowledge on the theory of valuations is…

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 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…

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…

We give model theoretic accounts and proofs of the existence and uniqueness of differential Galois extensions with no new constants, for logarithmic differential equations over a differential field K, when the field C of constants of K is…

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…

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…

We solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over ${\mathbb Q}$. More generally, we show that over such a field, every split differential…

If we consider a q-analogue of linear differential equation, Galoois group of the q-analogue difference equation is still a linear algebraic group. Namely, by a quantization of linear differential equation, Galois group is not quantized. We…

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…

We present a differential algebra of generalized functions over a field of generalized scalars by means of several axioms in terms of general algebra and topology. Our differential algebra is of Colombeau type in the sense that it contains…

We use Galois descent to construct central extensions of twisted forms of split simple Lie algebras over rings. These types of algebras arise naturally in the construction of Extended Affine Lie Algebras. The construction also gives…

Given a number field $k$, we show that, for many finite groups $G$, all the Galois extensions of $k$ with Galois group $G$ cannot be obtained by specializing any given finitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$…

We carry out some of Galois's work in the setting of an arbitrary first-order theory T. We replace the ambient algebraically closed field by a large model M of T, replace fields by definably closed subsets of M, assume that T codes finite…

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…

We give sufficient conditions for a linear differential equation to have a given semisimple group as its Galois group. For any linear algebraic group G given as a semidirect product of a finite subgroup and a normal subgroup that is a…

In this paper, we generalize the definition of the differential Galois group and the Galois correspondence theorem established previously for Picard-Vessiot extensions of real differential fields with real closed field of constants to any…