From Galois to Hopf Galois: Theory and Practice

Abstract

Hopf Galois theory expands the classical Galois theory by con- sidering 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 explic it 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 Ga lois module theory for wildly ramified extensions