Bounds on the number of ideals in finite commutative nilpotent \mathbbF_p-algebras

2017/06/08 by Childs, Lindsay N., Greither, Cornelius
#12F10 #13B05 (Secondary) #13M05 (Primary) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1706.02518

Abstract

Let A be a finite commutative nilpotent \mathbbF_p-algebra structure on G, an elementary abelian group of order pⁿ. If K/k is a Galois extension of fields with Galois group G and A^p = 0, then corresponding to A is an H-Hopf Galois structure on K/k of type G. For that Hopf Galois structure we may study the image of the Galois correspondence from k-subHopf algebras of H to subfields of K containing k by utilizing the fact that the intermediate subfields correspond to the \mathbbF_p-subspaces of A, while the subHopf algebras of H correspond to the ideals of A. We obtain upper and lower bounds on the proportion of subspaces of A that are ideals of A, and test the bounds on some examples.

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