Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bounds on the number of ideals in finite commutative nilpotent
  $\mathbb{F}_p$-algebras   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.

17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom
16
The categories of this paper are: stat
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc