Minimal Hopf-Galois Structures on Separable Field Extensions


In Hopf-Galois theory, every H-Hopf-Galois structure on a field extension K/k gives rise to an injective map \mathcal{F} from the set of k-sub-Hopf algebras of H into the intermediate fields of K/k. Recent papers on the failure of the surjectivity of \mathcal{F} reveal that there exist many Hopf-Galois structures for which there are many more subfields than sub-Hopf algebras. In this paper we survey and illustrate group-theoretical methods to determine H-Hopf-Galois structures on finite separable extensions in the extreme situation when H has only two sub-Hopf algebras. This corresponds to the case when the lack of surjectivity is at its extreme.

DOI Code: 10.1285/i15900932v41n1p55

Keywords: Galois and Hopf-Galois field extensions; Galois correspondence; characteristically simple groups

Full Text: PDF

Creative Commons License
This work is licensed under a Creative Commons Attribuzione - Non commerciale - Non opere derivate 3.0 Italia License.