ABSTRACT
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. To accomplish this goal, the section first considers analogues for the
concepts of irreducible, prime and associate. The ultimately insignificant distinction between two elements in a ring that are associates of one another leads to the definition of a principal ideal in a commutative ring R: 〈a〉 = {b ∈ R : b = ac, c ∈ R}. Statements about elements can be efficiently and elegantly translated into statements about principal ideals, especially for integral domains D (Theorem 13.3): 〈a〉 = D if and only if a is a unit; 〈a〉 ⊆ 〈b〉 if and only a divides b; a is irreducible if and only if 〈a〉 is maximal among all principal ideals. The idea of principal ideal can be generalized to that of ideal, which
is a subring I of a commutative ring R satisfying the multiplicative absorption property: if a ∈ I and r ∈ R, then ar ∈ I. For Z and Q[x], all ideals are principal (Theorem 11.6). An integral
domain where this holds is called a principal ideal domain (or PID). For a PID, we have factorization into irreducibles (Theorem 12.1). To prove uniqueness of such factorization requires precisely that the concepts of irreducibility and primeness coincide (Theorems 13.4 and 13.5), as they do for the integers. An integral domain that has unique factorization into irreducibles is called a unique factorization domains (or UFD). Integral domains of the form Z[
√ n] = {a+ b√n : a, b ∈ Z} are called
quadratic extensions of the integers. These are important examples; some such rings are PIDs, but not all. The ring Z[
√−5] is not a PID and is not even a UFD. The ring Z[x] of polynomials with integer coefficients is not a PID,
but it is a UFD; this follows essentially from Gauss’s Lemma. Some quadratic extensions (such as the Gaussian integers Z[
√−1]) share even more properties in common with Z and Q[x] and are called Euclidean domains: elements in such domains have a notion of ‘size’, which equips them with a Division algorithm. This makes it easy to prove (Theorem 15.3) that they are PIDs, and hence UFDs.
