Galois Theory: Solvability and Radical Extensions

This lecture covers the concept of solvability by radicals in Galois theory, where a polynomial equation is solvable if its roots can be expressed in terms of radicals. It also discusses basic and radical extensions, radical towers, and the Galois/Abel criterion for solvability. The instructor explains the conditions under which an extension is considered solvable and the implications for the group structure. Additionally, the lecture explores the relationship between the splitting field of an irreducible polynomial and its solvability by radicals, providing insights into the fundamental principles of Galois theory.