The Hopf Galois property in subfield lattices

Hopf Galois theory for finite separable field extensions was introduced by Greither and Pareigis. They showed that all Hopf Galois extensions of degree up to 5 are either Galois or almost classically Galois and they determined the Hopf Galois character of a separable extension according to the Galois group (or the degree) of its Galois closure. In this paper we study degree 6 separable extensions as well as intermediate extensions for degrees 4,5 and 6. We present an example of a non almost classically Galois Hopf Galois extension of the field of rational numbers of the smallest possible degree and new examples of Hopf Galois extensions. In the last section we prove a transitivity property of the Hopf Galois condition.