Definition 17.4.1 (Unique factorization domain). An integral domain is a unique factorization domain (UFD) if:
Every nonzero non-unit can be written as a product of irreducibles: .
This factorization is unique up to order and associates: if , then and (after reordering) each is an associate of .
Theorem 17.4.1 (Every PID is a UFD). Every principal ideal domain is a unique factorization domain.
Proof (sketch). Existence of factorization follows from the ascending chain condition on principal ideals in a PID: if is not irreducible, write with neither nor a unit. Then and . Repeat for and . This process terminates because an infinite strictly ascending chain would generate an ideal that is not principal, contradicting the PID hypothesis.
Uniqueness follows from the fact that irreducibles are prime in a PID: if , then , so for some ; since is irreducible, and are associates. Cancel and proceed by induction. ∎
The chain of implications is: None of these implications reverse.
Example 17.4.1 (Failure of unique factorization). In , we have . The norm is multiplicative: . Since ,,, and no element has norm 2 or 3, each of these four elements is irreducible. But and are not associates (they have different norms). So is not a UFD.