Let L/K be a Galois extension of fields with Galois group G an elementary abelian p-group of rank n for p an odd prime. It is known that nilpotent F_p-algebra structures A on G yield regular subgroups of the holomorph Hol(G), hence Hopf Galois structures on L/K. In this paper we illustrate the richness of Hopf Galois structures on L/K by examining the case where A is abelian of F_p-dimension n where dim(A²) = 1. We determine the number of Hopf Galois structures that arise in these cases, describe those structures explicitly, and estimate the extent of failure of surjectivity of the Galois correspondence for those structures.

This research was inspired by discussions with Tim Kohl. Many thanks to him for sharing his enthusiasm with me. My thanks also to the referee for some insightful comments.