Galois Theory: Extensions and Residual Fields

This lecture covers the Galois theory of field extensions, focusing on the properties of Galois extensions and unramified primes. It discusses the isomorphism induced by Galois extensions, the unramified condition for primes, and the behavior of roots of polynomials over field extensions. Additionally, it explores the concept of finite residual extensions, separability, and the Frobenius element at unramified primes. The lecture provides examples and definitions to clarify the theoretical concepts presented.