Explore the foundations of Galois theory in this 44-minute math history lecture. Delve into the classical problem of solving polynomial equations using radicals, tracing its historical development from ancient quadratic solutions to 16th-century cubic breakthroughs. Examine the quartic equation, factoring methods, and the relationship between roots and coefficients. Discover symmetric functions, Newton's identities, and Lagrange's pivotal contributions to modern algebraic approaches. Learn about the significance of symmetries in equation solving, the concept of resolvents, and the groundbreaking work of Ruffini and Abel that paved the way for Galois' revolutionary insights.