Dive into the world of polynomial rings in this comprehensive 2.5-hour lecture on abstract algebra. Explore key concepts such as the division algorithm for polynomials, techniques for expressing the greatest common divisor (gcd) of polynomials as a combination, and the properties of irreducible polynomials. Learn about Eisenstein's criterion and its applications, work through practical examples of writing polynomial gcds as combinations, and discover how to construct a field of order 4. Conclude by understanding why k[x] is a Principal Ideal Domain (PID), gaining a deeper appreciation for the fundamental structures in abstract algebra.