 

Lindsay N. Childs
On Abelian Hopf Galois structures and finite commutative nilpotent rings view print


Published: 
April 12, 2015

Keywords: 
Finite commutative nilpotent algebras, Hopf Galois extensions of fields, regular subgroups of finite affine groups 
Subject: 
Primary: 13E10, 12F10; Secondary: 20B35 


Abstract
Let G be an elementary abelian pgroup of rank n, with p an odd prime. In order to count the Hopf Galois structures of type G on a Galois extension of fields with Galois group G, we need to determine the orbits under conjugation by Aut(G) of regular subgroups of the holomorph of G that are isomorphic to G. The orbits correspond to isomorphism types of commutative nilpotent F_{p}algebras N of dimension n with N^{p} = 0. Adapting arguments of Kruse and Price, we obtain lower and upper bounds on the number f_{c}(n) of isomorphism types of commutative nilpotent algebras N of dimension n (as vector spaces) over the field F_{p} satisfying N^{3} = 0. For n = 3, 4 there are five, resp. eleven isomorphism types of commutative nilpotent algebras, independent of p (for p > 3). For n ≧ 6, we show that f_{c}(n) depends on p. In particular, for n = 6 we show that f_{c}(n) ≧ \lfloor (p1)/6 \rfloor by adapting an argument of Suprunenko and Tyschkevich. For n ≧ 7, f_{c}(n) ≧ p^{n6}. Conjecturally, f_{c}(5) is finite and independent of p, but that case remains open. Finally, applying a result of Poonen, we observe that the number of Hopf Galois structures of type G is asymptotic to f_{c}(n) as n goes to infinity.


Author information
Department of Mathematics and Statistics, University at Albany, Albany, NY 12222
lchilds@albany.edu

