We next show that dd i this follows immediately from

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We next show that dD I . This follows immediately from the fact that d I and the closure properties of ideals. That proves the existence part of the theorem. As for uniqueness, note that if dD = d 0 D , we have d | d 0 and d 0 | d , from which it follows that d 0 = ud for a unit u . 2 For a, b D , we call d D a common divisor of a and b if d | a and d | b ; moreover, we call d the greatest common divisor of a and b if d is normalized, and all other common divisors of a and b divide d . It is immediate from the definition of a greatest common divisor that it is unique if it exists at all. Analogous to Theorem 1.5, we have: Theorem 6.3 For any a, b D , there exists a greatest common divisor d of a and b , and moreover, aD + bD = dD ; in particular, as + bt = d for some s, t D . Proof. Replace the symbol Z in the proof of Theorem 1.5 with the symbol D . 2 For a, b D , we denote by gcd( a, b ) the greatest common divisor of a and b . We say that a and b are relatively prime if gcd( a, b ) = 1. Notice that a and b are relatively prime if and only if aD + bD = D , i.e., if and only if there exist s, t D such that as + bt = 1. Analogous to Theorem 1.6, we have: Theorem 6.4 For a, b, c D such that c | ab and gcd( a, c ) = 1 , we have c | b . Proof. Replace the symbol Z in the proof of Theorem 1.6 with the symbol D . 2 Analogous to Theorem 1.7, we have: Theorem 6.5 Let p D be irreducible, and let a, b D . Then p | ab implies that p | a or p | b . Proof. The only divisors of p are associate to 1 or p . Thus, gcd( p, a ) is either 1 or the monic associate of p . If p | a , we are done; otherwise, if p - a , we must have gcd( p, a ) = 1, and by the previous theorem, we conclude that p | b . 2 45
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Now to prove the uniqueness part of Theorem 6.1. Clearly, the choice of the unit u is uniquely determined: u = lc( n ). Suppose we have p 1 · · · p r = p 0 1 · · · p 0 s , where the p i and p 0 i are monic irreducible polynomials (duplicates are allowed among the p i and among the p 0 i ). If r = 0, we must have s = 0 and we are done. Otherwise, as p 1 divides the right-hand side, by inductively applying Theorem 6.5, one sees that p 1 is equal to some p 0 i . We can cancel these terms and proceed inductively (on r ). That completes the proof of Theorem 6.1. For non-zero polynomials a and b , it is easy to see that gcd( a, b ) = Y p p min( ν p ( a ) p ( b )) , where the function ν p ( · ) is as implicitly defined in Theorem 6.1. For a, b D a common multiple of a and b is a polynomial m such that a | m and b | m ; moreover, m is a least common multiple of a and b if m is normalized, and m divides all common multiples of a and b . In light of Theorem 6.1, it is clear that the least common multiple exists and is unique; indeed, if we denote the least common multiple of a and b as lcm( a, b ), then for non-zero polynomials a and b , we have lcm( a, b ) = Y p p max( ν p ( a ) p ( b )) . Moreover, for all a, b D , we have gcd( a, b ) · lcm( a, b ) = ab.
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