Counting Hopf Galois Structures on Non-Abelian Galois Field Extensions

Abstract

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.) The observation of Greither and Pareigis is the only one in the literature to this point which gives any information on the number of Hopf Galois structures on Galois field extensions for G non-abelian. The purpose of this paper is to make a start at determining the number of Hopf Galois structures on L/K for some non-abelian Galois groups G. Before stating our results, we need to describe Byott's counting formula.

Cayley's Theorem and Hopf Galois structures for semidirect products of cyclic groups

“…However, if we consider embeddings of G into InHol(G) = G Inn(G), then understanding regularity is made easier by the following observations, first utilized in [CC99]:…”

Hopf–Galois structures on Galois field extensions of degree pq

“…As a consequence of the work of Kohl [8] it is known that, for an odd prime p, there are p m−1 Hopf-Galois structures on a cyclic extension of degree p m . Carnahan and Childs [3] have shown that when G is the symmetric group S n (with n ¿ 5), there are at least (n!) 1=2 Hopf-Galois structures.…”

Hopf-Galois structures on a Galois S-extension

“…Remark 1.2. The first and last cases in Theorem 1.1 were known previously, by [5,Theorem 7] and [7,Proposition 4], respectively.…”