From Galois to Hopf Galois: theory and practice

TL;DR

This paper reviews the extension of classical Galois theory to Hopf Galois theory, focusing on separable extensions, their classification, explicit computations, and applications in Galois module theory for ramified extensions.

Contribution

It provides a comprehensive overview of Hopf Galois theory, including group-theoretical classification and explicit computational methods for separable extensions.

Findings

Classification of Hopf Galois structures for separable extensions

Explicit methods for computing Hopf Galois structures

Application to Galois module theory in ramified extensions

Abstract

Hopf Galois theory expands the classical Galois theory by considering the Galois property in terms of the action of the group algebra k[G] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable extensions where the Hopf Galois property admits a group-theoretical formulation suitable for counting and classifying, and also to perform explicit computations and explicit descriptions of all the ingredients involved in a Hopf Galois structure. At the end we give just a glimpse of how this theory is used in the context of Galois module theory for wildly ramified extensions.