Hopf-Galois Extensions and Applications

Explore the fascinating world of Hopf-Galois theory in this 55-minute lecture presented by Tony Ezome from École Normale Supérieure and PREMA. Delve into the generalization of classical Galois theory, tracing its origins from Chase and Sweedler's 1969 introduction to Greither and Pareigis' 1987 development for separable field extensions. Examine minimal Hopf-Galois structures of separable field extensions, investigate normal bases of Hopf-Galois extensions constructed from algebraic curves, and uncover the resulting arithmetic properties. Gain insights into this powerful mathematical framework that has applications in studying purely inseparable extensions of fields and ramified extensions of rings.