ABSTRACT
Section VII in a Nutshell This section examines conditions that Z and Q[x] share, which provide them with a unique factorization theorem into irreducibles, like the Fundamental Theorem of Arithmetic.
For Z and Q[x], all ideals are principal. An integral domain where this holds is called a principal ideal domain (or PID). For a PID, we have factorization into irreducibles (Theorem 32.2). To prove uniqueness of such factorization requires precisely that the concepts of irreducibility and primeness coincide (Theorems 32.4), as they do for the integers. An integral domain that has unique factorization into irreducibles is called a unique factorization domain (or UFD).
