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13. Integral Domains and Fields 1 Integral Domains While the set of integers is one of our prototypical examples of a ring, there are too many important properties of Z unaccounted for in the definition of ring; besides being commutative and having a unity element, the most important arithmetical property of Z not captured by satisfying the definition of a ring is that while only 1 and –1 are units, there is still a cancellation property: ab = ac ⇒ b = c . This leads to the following definitions. The element a in a ring is called a zero divisor if there exists a nonzero b in the ring so that ab = 0. € A commutative ring with unity is called an integral domain if it contains no zero divisors. € Examples: • Z is an integral domain (of course!) • Zn is an integral domain only when n is a prime, for if n = ab is a nontrivial factorization of n, then ab = 0 in this ring • Z[x] is an integral domain 13. Integral Domains and Fields 2 Theorem If a, b, and c are elements of an integral domain D and a ≠ 0, then ab = ac ⇒ b = c . Proof ab = ac ⇒ a(b − c) = 0 ⇒ b − c = 0 since b – c cannot be a zero divisor. // € € In much the same way that the structure of an integral domain is more descriptive of the integers than the basic structure of a ring, the rings Q, R, and C all share a basic property not identified by the fact that they are rings: in each example, every nonzero element is a unit. Any ring in which all nonzero elements are units is called a field. There are other rings that have the additional structure of a field. Theorem Any finite integral domain is a field. Proof Let D be a finite integral domain and suppose that a is any nonzero element. If a = 1, then a is its own inverse. If not, the list of powers of a must eventually repeat: there are positive integers i > j so that a i = a j . By cancellation, we get a i− j = 1. But a ≠ 1, so i – j > 1 and the inverse of a is a i− j −1. // € € € 13. Integral Domains and Fields 3 Corollary If p is a prime, then Z p is a field. Proof All we need to show is that Z p contains no zero divisors. So suppose € ab mod p = 0. Then there is some integer k so that ab = pk, whence p divdes the product ab. It follows that either p divides a (a mod p = 0) or p divides b€(b mod p = 0). So neither a nor b is a zero divisor. // € € € € More examples of fields: • Z3 [i], the set of all polynomial expressions in powers of i = −1 with coefficients from the field Z3 (the Gaussian integers mod 3); any element of this ring has the form a + bi with a,b ∈ Z3 since i 2€= −1 ∈ Z3 , and every nonzero element is a unit (see the multiplication table on p. 251) • Q[ 2], the set of polynomial € expressions in powers of 2 with rational coefficients; again, any element of this ring has the form a + b 2 for rational a and b, and all nonzero elements are units € because 1 a+b 2 € = 1 ⋅ a −b 2 a +b 2 a −b 2 = € a b − 2 2 2 2 2 a − 2b a − 2b 13. Integral Domains and Fields 4 The characteristic of a ring R, denoted char R, is the smallest positive integer n such that nx = 0 for all x in the ring; if no such integer exists, we say that the ring has characteristic 0. Theorem A ring R with unity 1 has positive characteristic n if and only if the order of 1 within the additive group that defines R equals n. The ring has characteristic 0 if and only if 1 has infinite order under addition. Proof 1 has additive order n > 0 ⇔ n ⋅1 = 0 ⇔ nx = x1+4 x2 + L+ 12 +L 4 44 3x = (11+4 4+1 3 )x = (n ⋅1)x = 0 ⇔ n terms n terms char R = n. Further, if 1 € has infinite order then char R must be 0; the converse is clear. // € 13. Integral Domains and Fields 5 Theorem If D is an integral domain, then char D is either 0 or a prime number. Proof If 1 has infinite order in D, then char D = 0. Otherwise, char D equals the finite order n of 1 under addition. But if n is composite, it factors as n = rs with 0 < r, s < n. So 0 = n ⋅1 = (rs) ⋅1 = r(s ⋅1) = (r ⋅1)(s ⋅1), and since there are no zero divisors, either r ⋅1 = 0 or s ⋅1 = 0, in violation of the fact that n is the € smallest positive integer so that n ⋅1 = 0. Thus n must be prime. // € € €