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

## TL;DR

This paper investigates the relationship between subspaces and ideals in finite commutative nilpotent $F_p$-algebras, providing bounds on the proportion of subspaces that are ideals, with applications to Hopf Galois structures.

## Contribution

It establishes bounds on the proportion of ideals among subspaces in finite commutative nilpotent $F_p$-algebras and explores their implications for Hopf Galois theory.

## Key findings

- Derived upper and lower bounds for the proportion of ideals among subspaces.
- Tested bounds on specific algebra examples.
- Connected algebraic structures to Galois correspondence in field extensions.

## 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.