— The paper consists of two parts. In the first part, we explain an excellent idea, due to mathematicians of the 19-th century, of naturally developing classical Galois theory of algebraic equations to an infinite dimensional Galois theory of nonlinear differential equations. We show with an instructive example how we can realize the idea of the 19-th century in a rigorous framework. In the sec...

Let L|K be a Galois extension of fields with finite Galois group G. Greither and Pareigis [GP87] showed that there is a bijection between Hopf Galois structures on L|K and regular subgroups of Perm(G) normalized by G, and Byott [By96] translated the problem into that of finding equivalence classes of embeddings of G in the holomorph of groups N of the same cardinality as G. In [CCo06] we showed...

We prove new cases of the inverse Galois problem by considering the residual Galois representations arising from a fixed newform. Specific choices of weight 3 newforms will show that there are Galois extensions of Q with Galois group PSL2(Fp) for all primes p and PSL2(Fp3) for all odd primes p ≡ ±2,±3,±4,±6 (mod 13).

Introduction. Galois extensions of noncommutative rings were introduced in 1964 by Teruo Kanzaki [13]. These algebraic objects generalize to noncommutative rings the classical Galois extensions of fields and the Galois extensions of commutative rings due to Auslander and Goldman [1]. At the same time they also turn out to be fundamental examples of Hopf-Galois extensions; these were first consi...

The classical Galois theory of fields and the classification of covering spaces of a path-connected, locally path-connected, and semi-locally simply connected space (which will be referred to as the Galois theory of covering spaces) appear very similar. We study the connection of these two Galois theories by generalizing them in categorical language as equivalences of certain categories. This i...

We study Galois points for a hypersurface X with dimSing(X) ≤ dimX − 2. The purpose of this article is to determine the set ∆(X) of Galois points in characteristic zero: Indeed, we give a sharp upper bound of the number of Galois points in terms of dimX and dimSing(X) if ∆(X) is a finite set, and prove that X is a cone if ∆(X) is infinite. To achieve our purpose, we need a certain hyperplane se...

Galois theory translates questions about fields into questions about groups. The fundamental theorem of Galois theory states that there is a bijection between the intermediate fields of a field extension and the subgroups of the corresponding Galois group. After a basic introduction to category and Galois theory, this project recasts the fundamental theorem of Galois theory using categorical la...

This paper introduces basic Galois Theory, primarily over fields with characteristic 0, beginning with polynomials and fields and ultimately relating the two with the Fundamental Theorem of Galois Theory. This paper then applies Galois Theory to prove Galois’s Theorem, describing the relationship between the Galois groups of polynomials and their solvability by radicals.

We survey some constructive aspects of di erential Galois theory and indicate some analogies between ordinary Galois theory and di erential Galois theory in characteristic zero and nonzero.

A locally connected topos is a Galois topos if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application of the construction of 2-filtered bilimits of topoi, we show that every Galois topos has a point. introduction. Galois topoi (definition 1.5) arise in Grothe...