In this note we study Morita contexts and Galois extensions for corings. For a coring C over a (not necessarily commutative) ground ring A we give equivalent conditions for M to satisfy the weak. resp. the strong structure theorem. We also characterize the so called cleft C-Galois extensions over commutative rings. Our approach is similar to that of Y. Doi and A. Masuoka in their work on (cleft...

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...

let $f$ be a finite extension of $bbb q$‎, ‎${bbb q}_p$ or a global‎ ‎field of positive characteristic‎, ‎and let $e/f$ be a galois extension‎. ‎we study the realization fields of‎ ‎finite subgroups $g$ of $gl_n(e)$ stable under the natural‎ ‎operation of the galois group of $e/f$‎. ‎though for sufficiently large $n$ and a fixed‎ algebraic number field $f$ every its finite extension $e$ is‎ ‎re...

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...

We find a relationship between regular embeddings of G, an elementary abelian p-group of order p, into unipotent upper triangular matrices with entries in Fp and commutative dimension n degree 2 polynomial formal groups with nilpotent upper triangular structure matrices. We classify the latter when n = 3 up to linear isomorphism, and use that classification to determine the number of Hopf Galoi...

This thesis is concerned with the definition and the study of properties of homotopic Hopf-Galois extensions in the category Ch 0 k of chain complexes over a field k, equipped with its projective model structure. Given a differential graded k-Hopf algebra H of finite type, we define a homotopic H-Hopf-Galois extension to be a morphism ' : B ! A of augmented H-comodule dg-k-algebras, where B is ...