The Hopf Galois Property in Subfield Lattices

Let K/k be a finite separable extension, n its degree and (K) over tilde /k its Galois closure. For n <= 5, Greither and Pareigis show that all Hopf Galois extensions are either Galois or almost classically Galois and they determine the Hopf Galois character of K/ k according to the Galois group (or the degree) of (K) over tilde /k. In this paper we study the case n = 6, and intermediate extensions F/ k such that K subset of F subset of (K) over tilde, for degrees n = 4, 5, 6. We present an example of a non almost classically Galois Hopf Galois extension of (sic) 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.