| dc.description.abstract | Hopf-Galois structures, originally due to Chase and Sweedler, generalise the concept of a Galois extension of fields. In contrast to classical Galois theory, which associates a unique Galois group to each Galois extension of fields, there can be multiple Hopf-Galois structures on a given field extension. It is therefore of interest to understand the types of these Hopf-Galois structures, and how many can exist.
Existing work has already been conducted on a variety of classes of extensions, in particular the case of a cyclic extensions of squarefree and of prime-power degree have been studied, as well as the structures on extensions of degree pq for two primes with p ≡ 1 (mod q). In many remaining cases, however, the total number of these structures is an open question. We seek an enumeration of the Hopf-Galois structures on cyclic Galois extensions of arbitrary degree n. In particular we obtain strict conditions on the permissible types of such structures, showing that if G is the type of a structure then G has a characteristic subgroup of index a power of 2. We then classify the possible types in terms of the 2-Sylow subgroups. To obtain an enumeration we perform group theoretic calculations, to find regular cyclic subgroups of the holomorph Hol(G). Then, correcting for conjugate subgroups, we obtain a complete count of the Hopf-Galois structures of type G, in the cases where the 2-Sylow subgroup of G is cyclic, dihedral, or the elementary abelian group C2 × C2. The only other possibility is that the 2-Sylow subgroup of G is a generalised quaternion group of order 2n for n ≥ 4, which is not treated in this thesis. | en_GB |
