25 Unique Factorization Domain

In this lecture, we revisit the concept of the Unique Factorization Domain (UFD) and study it in greater depth, building connections with classical results in algebra. Definition: We begin by formally defining a Unique Factorization Domain (UFD), a ring in which every non-zero, non-unit element can be written as a product of irreducible elements, and this factorization is unique up to order and associates. Key Results Discussed: ℤ as a UFD: We demonstrate that the ring of integers ℤ is a unique factorization domain, using the Fundamental Theorem of Arithmetic. PIDs are UFDs: We prove that every Principal Ideal Domain (PID) is a UFD, establishing a powerful structural result. Polynomial Rings: For any field F, the polynomial ring F[x] is a Unique Factorization Domain. Eisenstein’s Criterion (Revisited): We present another proof of Eisenstein’s Criterion for irreducibility, this time using the framework of UFDs. This perspective is essential for deeper understanding and future applications. This lecture strengthens the foundation of factorization theory, bridging the ideas of irreducibles, primes, PIDs, and UFDs, and prepares us for more advanced topics in algebra. 👉 Keep moving forward in your Ring Theory journey! Subscribe now and explore the playlists of whole 5th Semester (B.Sc. Hons Mathematics, DU) for step-by-step, exam-oriented lectures. 📌 Watch the full Ring Theory playlist here: https://www.youtube.com/playlist?list=PLzt330quwYmVX9V6_fyUEo-kH94VizR1t Android App Download Link: https://play.google.com/store/apps/details?id=com.ynpwie.dswxqw Windows App Download Link: https://appxcontent.kaxa.in/windows/The_ClassRoom_Study_Setup_0.0.1.exe Website Link: https://theclassroomstudy.akamai.net.in/ iOS App Download Link: https://apps.apple.com/in/app/my-appx/id1662307591 (Use Organization ID: 4234816)