Abstract Algebra II: Rings, Fields, and Galois Theory

Abstract algebra provides the mathematical language underlying modern physics, computer science, and secure communications. This course offers a clear, step-by-step pathway to understanding advanced algebraic systems, transforming how you analyze abstract structures. You will start with the core definitions of rings and fields, building a solid foundation before moving on to complex concepts like factorization and field extensions. By the end of this text-only course, you will be able to confidently read, write, and analyze algebraic proofs and understand how these theoretical concepts apply to modern cryptography. What you'll learn: 1. Understand the foundational properties of rings, ideals, and homomorphisms. 2. Analyze polynomial rings and the mechanics of unique factorization domains. 3. Explore field extensions, splitting fields, and the core concepts of Galois theory. 4. Apply algebraic structures to conceptual frameworks in cryptography and coding theory. 5. Practice mathematical reasoning through detailed written proofs and step-by-step exercises. The course begins with essential terminology and fundamental definitions of rings, ensuring you have a strong starting point. From there, the written modules guide you through quotient rings, field properties, and field extensions, concluding with practical connections to modern computer science. This course is designed for undergraduate students, self-taught mathematicians, and computer science enthusiasts who have a basic familiarity with groups and want to expand their algebraic toolkit. No advanced mathematical background is required. Begin your journey into advanced mathematics and start reading today.