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...

Abstract Let $$\Gamma $$ Γ be a finite group acting on Lie G . We consider class of extensions $$1 \rightarrow \hat{G} \Gamma 1$$ 1 → G ^ defined by this action and 2-cocycle with values in the ...

Let L be a field which is a Galois extension of the field K with Galois group G. Greither and Pareigis [GP87] showed that for many G there exist K-Hopf algebras H other than the group ring KG which make L into an H-Hopf Galois extension of K (or a Galois H∗object in the sense of Chase and Sweedler [CS69]). Using Galois descent they translated the problem of determining the Hopf Galois structure...

(a) Find Galois points and the Galois groups for singular plane curves. – for smooth curves, the number of Galois points is at most three (resp. four) if they are outer (resp. inner). The Galois groups are cyclic. [46, 62] – (i) How is the structure of Galois group and how many Galois points do there exist? Is it true that the maximal number of outer (resp. inner) Galois points is three (resp. ...

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...

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.

We establish automatic realizations of Galois groups among groups M ⋊ G, where G is a cyclic group of order p for a prime p and M is a quotient of the group ring Fp[G]. The fundamental problem in inverse Galois theory is to determine, for a given field F and a given profinite group G, whether there exists a Galois extension K/F such that Gal(K/F ) is isomorphic to G. A natural sort of reduction...