On the Galois correspondence ratio for Hopf-Galois extensions arising from nilpotent ${\mathbb{F}}_p$-algebras

Abstract

For a Hopf-Galois structure on a Galois extension $L/K$ of fields that arises from a finite nilpotent ${\mathbb{F}}_p$-algebra $A$, we look at the Galois correspondence ratio, which measures the failure of surjectivity of the Galois correspondence for the Hopf-Galois structure on $L/K$. Using methods of elementary linear algebra, we observe that the number of subgroups of the adjoint group of $A$ is equal to the number of subgroups of the additive group of $N$. Then we count left ideals of $A$ and thereby determine the GCR for all nilpotent ${\mathbb{F}}_p$-algebras of dimension 4, and also show that for a set of ${\mathbb{F}}_p$-algebras of arbitrary dimension $n$ and exponent $e$, the GCR approaches 0 for large $p$, $n$ or $e$.

Theorem 1 is false for A = F_2[x]/(x^3): (A, +) \cong C_2 x C_2; (A, \circ) \cong C_4