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Hopf-Galois structures on separable field extensions of degree $pq$

Andrew Darlington

Abstract

In 2020, Alabdali and Byott described the Hopf-Galois structures arising on Galois field extensions of squarefree degree. Extending to squarefree separable, but not necessarily normal, extensions $L/K$ is a natural next step. One must consider now the interplay between two Galois groups $G=\operatorname{Gal}(E/K)$ and $G'=\operatorname{Gal}(E/L)$, where $E$ is the Galois closure of $L/K$. In this paper, we give a characterisation and enumeration of the Hopf-Galois structures arising on separable extensions of degree $pq$ where $p$ and $q$ are distinct odd primes. This work includes the results of Byott and Martin-Lyons who do likewise for the special case that $p=2q+1$.

Hopf-Galois structures on separable field extensions of degree $pq$

Abstract

In 2020, Alabdali and Byott described the Hopf-Galois structures arising on Galois field extensions of squarefree degree. Extending to squarefree separable, but not necessarily normal, extensions is a natural next step. One must consider now the interplay between two Galois groups and , where is the Galois closure of . In this paper, we give a characterisation and enumeration of the Hopf-Galois structures arising on separable extensions of degree where and are distinct odd primes. This work includes the results of Byott and Martin-Lyons who do likewise for the special case that .
Paper Structure (5 sections, 21 theorems, 88 equations, 6 tables)

This paper contains 5 sections, 21 theorems, 88 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hopf-Galois Structures for General $pq$
  4. Cyclic case
  5. Metabelian case

Key Result

Lemma 2.1

Let $G,G',N$ be as above, let $e(G,N)$ be the number of Hopf-Galois structures of type $N$ which realise $G$, and $e'(G,N)$ the number of subgroups $M$ of $\emph{Hol}(N)$ which are transitive on $N$ and isomorphic to $G$ via an isomorphism taking the stabiliser $M'$ of $1_N$ in $M$ to $G'$. Then where the group of automorphisms $\theta$ of $G$ such that $\theta$ fixes the identity coset $1_GG'$

Theorems & Definitions (41)

  • Lemma 2.1: Byo96
  • Proposition 2.2
  • Proposition 2.3
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 31 more