Counting Hopf Galois Structures on Non-abelian Galois Field Extensions

Let L be a field which is a Galois extension of the field K with Galois group G. Greither and Pareigis [GP87] showed that for many G there exist K-Hopf algebras H other than the group ring KG which make L into an H-Hopf Galois extension of K (or a Galois H∗object in the sense of Chase and Sweedler [CS69]). Using Galois descent they translated the problem of determining the Hopf Galois structures on L/K into one which depends only on the Galois group G. By this translation, they showed, for example, that any Galois extension with non-abelian G admits at least one non-classical Hopf Galois structure. Byott [By96] further translated the problem to a more amenable group-theoretic problem, and showed that a Galois extension L/K of fields with group G has a unique Hopf Galois structure, namely that by KG, iff n, the order of G, is a Burnside number, that is, is coprime to φ(n), Euler’s phi-function of n. (This implies that G is cyclic of square-free order.)