 

Lindsay N. Childs
Abelian Hopf Galois structures from almost trivial commutative nilpotent algebras
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Published: 
December 15, 2019. 
Keywords: 
Finite commutative nilpotent algebras, abelian Hopf Galois extensions of fields, regular subgroups of finite affine groups. 
Subject: 
13B05, 12F10, 20B35, 13M05. 


Abstract
Let L/K be a Galois extension of fields with Galois group G an elementary abelian pgroup 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(A^{2}) = 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. 

Acknowledgements
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.


Author information
Lindsay N. Childs:
Department of Mathematics and Statistics
University at Albany, SUNY
Albany, NY 12222, USA
lchilds@albany.edu

