Galois Theory Fundamentals

This lecture covers the fundamentals of Galois theory, focusing on separable elements, decomposition fields, and Galois groups. It explains the equivalence between separable extensions and algebraic closures, as well as the properties of finite degree extensions. The lecture also delves into the theory of Galois groups, demonstrating the importance of separable elements and the structure of Galois extensions. Additionally, it discusses the concept of cutting out elements, lemma on separable elements, and the generation of extensions by separable elements. The lecture concludes with examples illustrating the application of Galois theory in various fields.