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 extensions where the Hopf Galois property admits a group-theoretical formulation suitable for counting and classifying, and also to perform explicit computations an...

in a previous paper, the second author established that, given finite fields $f < e$ and certain subgroups $c leq e^times$, there is a galois connection between the intermediate field lattice ${l mid f leq l leq e}$ and $c$'s subgroup lattice. based on the galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product $c rtimes {rm gal} (e/f)$...

We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hi...

Let S, T be complete lattices. One can check that there exists a connection between S and T which is Galois. Next we state the proposition (1) Let S, T , S, T ′ be non empty relational structures. Suppose that (i) the relational structure of S = the relational structure of S, and (ii) the relational structure of T = the relational structure of T . Let c be a connection between S and T and c be ...

Higher dimensional central extensions of groups were introduced by G. Janelidze as particular instances of the abstract notion of covering morphism from categorical Galois theory. More recently, the notion has been extended to and studied in arbitrary semi-abelian categories. Here, we further extend the scope to exact Mal’tsev categories and beyond. For this, we consider conditions on a Galois ...

The concept of Galois connection between power sets is generalized from the point of view of fuzzy logic. Studied is the case where the structure of truth values forms a complete residuated lattice. It is proved that fuzzy Galois connections are in one-to-one correspondence with binary fuzzy relations. A representation of fuzzy Galois connections by (classical) Galois connections is provided. 1...

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 studies the structure of U(g)-Galois extensions. In particular, we use a result of Bell to construct a “PBW-like” free basis for faithfully flat U(g)-Galois extensions. We then move to non-faithfully flat extensions and propose a possible equivalent condition for a U(g)-extension to be Galois. We get a partial result for this.

Because the need to classify objects with respect to uncertain properties is increasing, it becomes important to seek an appropriate generalization of a Galois lattice structure. We aim to analyze an uncertain (probabilistic) Galois lattice structure. One of our interests consists to determine a complete (total) order within the objects set generated by the uncertain Galois lattice structure. A...