Hopf–Galois Structures, Skew Braces, and Their Connection

The study of Hopf–Galois structures, tools for addressing classical problems in arithmetic or field theory in a broader context, has been enriched by an unexpected connection with skew braces, algebraic structures whose relevance in various areas of mathematics has been demonstrated. Despite not being bijective, this connection has prompted the development of several quantitative results in recent years. However, the literature suggests that structural results have been relatively few. The primary goal of this dissertation is to propose a new version of the connection between Hopf–Galois structures and skew braces, aiming to render the connection bijective, explicit, and more structural. We prove that existing results can still be derived from this perspective while also uncovering new ones. Various structural properties of skew braces find concrete interpretations within the framework of Hopf–Galois structures; as a consequence, a longstanding problem in Hopf–Galois theory—the investigation of the bijectivity of the Hopf–Galois correspondence—can be translated into a problem in skew brace theory, and addressed in various cases. In the course of our analysis, we also investigate results concerning the algebraic structure of skew braces. For instance, we study the structural properties of certain subclasses, explore the relationship with Rota–Baxter operators via cohomological tools, and solve a classification problem posed by Vendramin.