On some semidirect products of skew braces arising in Hopf-Galois theory

TL;DR

This paper classifies certain skew braces related to Hopf-Galois structures in field extensions, providing a detailed understanding of their algebraic properties and implications for local field extensions.

Contribution

It introduces a classification of skew braces as semidirect products of ideals and applies this to classify Hopf-Galois structures in Galois extensions with specific group decompositions.

Findings

Classification of skew braces as semidirect products of ideals.

Explicit description of Hopf-Galois structures via smash products.

Analysis of normal basis generators and module structures in local field extensions.

Abstract

We classify skew braces that are the semidirect product of an ideal and a left ideal. As a consequence, given a Galois extension of fields $ L/K $ whose Galois group is the semidirect product of a normal subgroup $ A $ and a subgroup $ B $, we classify the Hopf-Galois structures on $ L/K $ that realize $ L^{A} $ via a normal Hopf subalgebra and $ L^{B} $ via a Hopf subalgebra. We show that the Hopf algebra giving such a Hopf-Galois structure is the smash product of these Hopf subalgebras, and use this description to study generalized normal basis generators and questions of integral module structure in extensions of local fields.