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.