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arXiv · 1706.02518

Bounds on the number of ideals in finite commutative nilpotent $\mathbb{F}_p$-algebras

Lindsay N. Childs·Cornelius Greither
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Abstract

Let $A$ be a finite commutative nilpotent $\mathbb{F}_p$-algebra structure on $G$, an elementary abelian group of order $p^n$. 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 $\mathbb{F}_p$-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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Lindsay N. Childs, Cornelius Greither. 2017-06-08. Bounds on the number of ideals in finite commutative nilpotent $\mathbb{F}_p$-algebras. https://arxiv.org/abs/1706.02518

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