What is a unique factorization domain?

A unique factorization domain is an integral domain where every nonzero, non-unit element can be expressed as a product of irreducible elements in a way that is unique up to order and units. This algebraic structure generalizes the fundamental theorem of arithmetic from ordinary integers to other rings, such as polynomial rings. Important examples include integers and polynomials with coefficients from a field or the integers.

What we know

• A unique factorization domain is an integral domain in which every nonzero non-unit element can be written as a product of irreducible elements uniquely up to order and units. [2]
• Important examples of unique factorization domains include the integers and polynomial rings in one or more variables with coefficients from the integers or a field. [2]
• Unique factorization domains appear in a chain of class inclusions between integral domains and principal ideal domains. [2]
• Certain rings of algebraic integers are examples of number systems that are not unique factorization domains. [4, 5]

Where to go next

• how we knowWhat is the fundamental theorem of arithmetic?
• the bigger pictureHow do principal ideal domains relate to UFDs?
• a case in pointWhy do algebraic integers fail unique factorization?
• compared withWhat is a noncommutative unique factorization domain?
• how it workedHow does polynomial factorization work in computer algebra?
• an unexpected connectionWhy did the failure of unique factorization stall Fermat's Last Theorem?The failure of unique factorization in rings of algebraic integers caused errors in many historical proofs attempted over centuries before Wiles successfully proved Fermat's Last Theorem.

Sources

• [1]unique factorization domain (Wikidata Q1052579) · CC0 1.0
• [2]Unique factorization domain · CC BY-SA 4.0
• [3]Noncommutative unique factorization domain · CC BY-SA 4.0
• [4]Fundamental theorem of arithmetic · CC BY-SA 4.0
• [5]Factorization · CC BY-SA 4.0