Minimal Hopf-Galois Structures on Separable Field Extensions

In Hopf-Galois theory, every -Hopf-Galois structure on a field extension gives rise to an injective map from the set of -sub-Hopf algebras of into the intermediate fields of . Recent papers on the failure of the surjectivity of 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 -Hopf-Galois structures on finite separable extensions in the extreme situation when has only two sub-Hopf algebras. This corresponds to the case when the lack of surjectivity is at its extreme.