• Hopf algebras
• Hopf Galois

We study the Hopf Galois theory for finite and separable fields extensions. Hopf Galois extensions can be thought of as a generalization of the concept of Galois extension, although they present some differences with the classical case; for example a given extension can have many different Hopf Galois structures. To know how many these structures are and how they look like, we discuss a remarkable theorem of Greither and Pareigis. We also present a Hopf Galois version of the correspondence theorem, due to Sweedler and Chase. For a special class of Hopf Galois extensions we get a stronger version of Swedleer and Chase's theorem and we propose a reformulation of this result, due to Crespo, Rio and Vela, that is more similar to the classical correspondence theorem. Finally, for extensions of high degree it becomes difficult to apply the aforementioned Greither and Pareigis theorem; a fundamental result by Byott allows us to reverse the question in such a way that we can work in a simpler setting. Following the work of Crespo, Rio and Vela, we apply Byott's theorem to classify the small degree extensions: we consider the cases degrees smaller than 8.