Abelian Hopf Galois structures from almost trivial commutative nilpotent algebras · arXivDesk
1604.05269Apr 18, 2016Substantially revised from the 2016 version, which was entitled "Obtaining abelian Hopf Galois structures from finite commutative nilpotent $\mathbb{F}_p$-algebras"
Abelian Hopf Galois structures from almost trivial commutative nilpotent algebras
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 Fp
Nearby in the stack
-algebra structures
A
on
G
yield regular subgroups of the holomorph of
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 dimension
n
where the dimension of
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.