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.

Non-isomorphic Hopf Galois structures with isomorphic underlying Hopf algebras

“…Within the class HG of separable Hopf Galois extensions, Greither and Pareigis defined the subclass AC of almost classically Galois extensions [7,Definition 4.2] and proved that AC is included in the class B of separable Hopf Galois extensions which may be endowed with a Hopf Galois structure such that the Galois correspondence is bijective [7,Theorem 5.2]. In [5], we proved that the two inclusions AC ⊂ B and B ⊂ HG are strict.…”

“…We shall now determine the image of the Galois correspondence for each of the Hopf Galois structures. To this end, we compute the subgroups of each of the corresponding N 's which are stable under conjugation by λ(G) (see the reformulation of the Galois correspondence theorem in terms of groups in [5,Theorem 2.3]). The action of τ and σ on the generators of each N i , 1 ≤ i ≤ 4, is given in the following table.…”

From Galois to Hopf Galois: Theory and Practice

“…But if all the subgroups of N are stable under conjugation by elements of λ(G) (as happened in the examples of the previous subsection) then we have injectivity. In fact, since N ′ and the stable subgroup σ∈G λ(σ)N ′ λ(σ) −1 give rise to the same sub-Hopf algebra, we have that sub-Hopf algebras of H are in bijection with subgroups of N stable under the action of λ(G) (see [16]). Therefore, the main theorem admits also a group-theoretical reformulation:…”

“…An example of degree 16 was constructed in [17], where the base field k is a quadratic extension of Q. In [16], we prove that the class of extensions for which the fundamental theorem of Galois theory holds in its strong form is larger than the class of almost classically Galois extensions by constructing a non-almost classically Galois extension for which the strong form holds.…”

Fundamentals of Hopf Algebras

“…For more examples of separable, non-Galois extensions of Q which have (or don't have) Hopf Galois structures on them, see [CRV14a,CRV14b] and [CRV14c].…”