Évariste Galois (1811–1832) has been increasingly recognised as an important mathematician who despite his short life developed mathematical ideas that today have led to applications in computer science (such as Galois connections) and elsewhere. Some of Galois’ mathematics can be visualised in interesting and even artistic ways, aided using software. In addition, a significant corpus of the hi...

In the study of Galois theory, after computing a few Galois groups of a given field, it is very natural to ask the question of whether or not every finite group can appear as a Galois group for that particular field. This question was first studied in depth by David Hilbert, and since then it has become known as the Inverse Galois Problem. It is usually posed as which groups appear as Galois ex...

Given a ring A and an A-coring C we study when the forgetful functor from the category of right C-comodules to the category of right A-modules and its right adjoint − ⊗A C are separable. We then proceed to study when the induction functor − ⊗A C is also the left adjoint of the forgetful functor. This question is closely related to the problem when A → AHom(C, A) is a Frobenius extension. We int...

Partial Galois extensions were recently introduced by Doku-chaev, Ferrero and Paques. We introduce partial Galois extensions for noncommutative rings, using the theory of Galois corings. We associate a Morita context to a partial action on a ring.

The smallest non-abelian p-groups play a fundamental role in the theory of Galois p-extensions. We illustrate this by highlighting their role in the definition of the norm residue map in Galois cohomology. We then determine how often these groups — as well as other closely related, larger p-groups — occur as Galois groups over given base fields. We show further how the appearance of some Galois...

The present information and technology processed by computers to access the private information like video conferencing, account status in bank, e – transaction of money and business deals through internet for all these encryption of the information in various forms are necessary. So to protect the information from different attacks using symmetric or asymmetric key encryption by adding the ran...

Two characterizations of an Azumaya Galois extension of a ring are given in terms of the Azumaya skew group ring of the Galois group over the extension and a Galois extension of a ring with a special Galois system is determined by the trace of the Galois group.

For a finite set S of primes of a global field K and for σ1, . . . , σe ∈ Gal(K) we denote the field of totally S-adic numbers by Ktot,S , the fixed field of σ1, . . . , σe in Ktot,S by Ktot,S(σ), and the maximal Galois extension of K in Ktot,S(σ) by Ktot,S [σ]. We prove that for almost all σ ∈ Gal(K) the absolute Galois group of Ktot,S [σ] is isomorphic to the free product of F̂ω and a free pro...

We present a detailed and simplified version of Hrushovski’s algorithm that determines the Galois group of a linear differential equation. There are three major ingredients in this algorithm. The first is to look for a degree bound for proto-Galois groups, which enables one to compute one of them. The second is to determine the identity component of the Galois group that is the pullback of a to...