376 research outputs found
From Galois to Hopf Galois: theory and practice
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 and explicit descriptions of all the ingredients involved in a
Hopf Galois structure. At the end we give just a glimpse of how this theory is
used in the context of Galois module theory for wildly ramified extensions
On Induced Hopf Galois Structures and their Local Hopf Galois Modules
The regular subgroup determining an induced Hopf Galois structure for a
Galois extension is obtained as direct product of the corresponding
regular groups of the inducing subextensions. We describe here the attached
Hopf algebra and Hopf action of an induced structure and we prove that they are
obtained by tensoring the corresponding inducing objects. We give a general
matrix description of the Hopf action which is useful to compute bases of
associated orders. In case of an induced Hopf Galois structures it allows us to
decompose the associated order, assuming that inducing subextensions are
arithmetically disjoint.Comment: Accepted to be published in Publicacions Matem\'atiques. We have
included two minor change
On the Galois correspondence theorem in separable Hopf Galois theory
In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that the image of the Galois correspondence does not determine the Hopf Galois structure
Inducing braces and Hopf Galois structures
Let be a prime number and let be an integer not divisible by and
such that every group of order has a normal subgroup of order . (This
holds in particular for .) We prove that left braces of size may be
obtained as a semidirect product of the unique left brace of size and a
left brace of size . We give a method to determine all braces of size
from the braces of size and certain classes of morphisms from the
multiplicative group of these braces of size to . From it
we derive a formula giving the number of Hopf Galois structures of abelian type
on a Galois extension of degree in terms of the
number of Hopf Galois structures of abelian type on a Galois extension of
degree . For a prime number , we apply the obtained results to
describe all left braces of size and determine the number of Hopf Galois
structures of abelian type on a Galois extension of degree .Comment: arXiv admin note: text overlap with arXiv:2205.0420
On the Galois correspondence theorem in separable Hopf Galois theory
In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that the image of the Galois correspondence does not determine the Hopf Galois structure
On the Galois correspondence theorem in separable Hopf Galois theory
Abstract: In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that the image of the Galois correspondence does not determine the Hopf Galois structure. 2010 Mathematics Subject Classification: Primary: 12F10; Secondary: 13B05, 16T05
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