Abstract Algebra Theory and Applications - Computer Science ...

16.2 FACTORIZATION IN INTEGRAL DOMAINS 287in Q(i). In the last steps we are writing the real and imaginary parts as aninteger plus a proper fraction. That is, we take the closest integer m i suchthat the fractional part satisfies |n i /(a 2 + b 2 )| ≤ 1/2. For example, we write98158= 1 + 1 8= 2 − 1 8 .Thus, s and t are the “fractional parts” of zw −1 = (m 1 + m 2 i) + (s + ti).We also know that s 2 + t 2 ≤ 1/4 + 1/4 = 1/2. Multiplying by w, we havez = zw −1 w = w(m 1 + m 2 i) + w(s + ti) = qw + r,where q = m 1 +m 2 i and r = w(s+ti). Since z and qw are in Z[i], r must bein Z[i]. Finally, we need to show that either r = 0 or ν(r) < ν(w). However,ν(r) = ν(w)ν(s + ti) ≤ 1 ν(w) < ν(w).2Theorem 16.13 Every Euclidean domain is a principal ideal domain.Proof. Let D be a Euclidean domain and let ν be a Euclidean valuationon D. Suppose I is a nontrivial ideal in D and choose a nonzero elementb ∈ I such that ν(b) is minimal for all a ∈ I. Since D is a Euclidean domain,there exist elements q and r in D such that a = bq + r and either r = 0 orν(r) < ν(b). But r = a − bq is in I since I is an ideal; therefore, r = 0 bythe minimality of b. It follows that a = bq and I = 〈b〉.□Corollary 16.14 Every Euclidean domain is a unique factorization domain.Factorization in D[x]One of the most important polynomial rings is Z[x]. One of the first questionsthat come to mind about Z[x] is whether or not it is a UFD. We willprove a more general statement here. Our first task is to obtain a moregeneral version of Gauss’s Lemma (Theorem 15.9).Let D be a unique factorization domain and suppose thatp(x) = a n x n + · · · + a 1 x + a 0