Abstract Algebra Integral Domains

Dive deep into the world of abstract algebra with a comprehensive exploration of integral domains. Learn about the field of fractions, irreducibles and primes, unique factorization domains (UFDs), principal ideal domains (PIDs), and Euclidean domains. Explore the relationships between these structures, including the proof that every PID is a UFD. Investigate the properties of polynomial rings over UFDs and examine specific examples of PIDs that are not Euclidean domains. Conclude with a study of ideals in quotients of PIDs, gaining a thorough understanding of these fundamental algebraic structures and their interconnections.

Syllabus

Abstract Algebra | The field of fractions of an integral domain..
Abstract Algebra | Irreducibles and Primes in Integral Domains.
Abstract Algebra | Introduction to Unique Factorization Domains.
Abstract Algebra | Introduction to Principal Ideal Domains (PIDs).
Abstract Algebra | Every PID is a UFD..
Abstract Algebra | Introduction to Euclidean Domains.
Abstract Algebra | If D is a UFD then D[x] is a UFD..
Abstract Algebra | Summary of Integral Domains.
Abstract Algebra | A PID that is not a Euclidean Domain.
Abstract Algebra | A PID that is not a Euclidean Domain.
Abstract Algebra | Ideals of quotients of PIDs.