Abstract Algebra Theory and Applications - Computer Science ...

286 CHAPTER 16 INTEGRAL DOMAINSEuclidean DomainsWe have repeatedly used the division algorithm when proving results abouteither Z or F [x], where F is a field. We should now ask when a divisionalgorithm is available for an integral domain.Let D be an integral domain such that for each a ∈ D there is a nonnegativeinteger ν(a) satisfying the following conditions.1. If a and b are nonzero elements in D, then ν(a) ≤ ν(ab).2. Let a, b ∈ D and suppose that b ≠ 0. Then there exist elementsq, r ∈ D such that a = bq + r and either r = 0 or ν(r) < ν(b).Then D is called a Euclidean domain and ν is called a Euclidean valuation.Example 6. Absolute value on Z is a Euclidean valuation.Example 7. Let F be a field. Then the degree of a polynomial in F [x] isa Euclidean valuation.Example 8. Recall that the Gaussian integers in Example 9 of Chapter 14are defined byZ[i] = {a + bi : a, b ∈ Z}.We usually measure the size of a complex number a + bi by its absolutevalue, |a + bi| = √ a 2 + b 2 ; however, √ a 2 + b 2 may not be an integer. Forour valuation we will let ν(a+bi) = a 2 +b 2 to ensure that we have an integer.We claim that ν(a + bi) = a 2 + b 2 is a Euclidean valuation on Z[i]. Letz, w ∈ Z[i]. Then ν(zw) = |zw| 2 = |z| 2 |w| 2 = ν(z)ν(w). Since ν(z) ≥ 1 forevery nonzero z ∈ Z[i], ν(z) = ν(z)ν(w).Next, we must show that for any z = a + bi and w = c + di in Z[i]with w ≠ 0, there exist elements q and r in Z[i] such that z = qw + rwith either r = 0 or ν(r) < ν(w). We can view z and w as elements inQ(i) = {p + qi : p, q ∈ Q}, the field of fractions of Z[i]. Observe thatzw −1 = (a + bi) c − dic 2 + d 2ac + bd bc − ad=c 2 ++ d2 c 2 + d 2 i(= m 1 + n ) (1c 2 + d 2 + m 2 + n )2c 2 + d 2 i(n1= (m 1 + m 2 i) +c 2 + d 2 + n )2c 2 + d 2 i= (m 1 + m 2 i) + (s + ti)