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Unformatted text preview: 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 ) ,ν 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 ) ,ν p ( b )) . Moreover, for all a,b ∈ D , we have gcd( a,b ) · lcm( a,b ) = ab. Recall that for polynomials a,b,n , we write a ≡ b (mod n ) when n | ( a- b ). For a non-zero polynomial n , and a ∈ D , we say that a is a unit modulo n if there exists a ∈ D such that aa ≡ 1 (mod n ), in which case we say that a is a multiplicative inverse of a modulo n . All of the results we proved in Chapter 2 for integer congruences carry over almost identically to polynomials. As such, we do not give proofs of any of the results here. The reader may simply check that the proofs of the corresponding results translate almost directly. Theorem 6.6 An polynomial a is a unit modulo n if and only if a and n are relatively prime. Theorem 6.7 If a is relatively prime to n , then ax ≡ ax (mod n ) if and only if x ≡ x (mod n ) . More generally, if d = gcd( a,n ) , then ax ≡ ax (mod n ) if and only if x ≡ x (mod n/d ) . Theorem 6.8 Let n be a non-zero polynomial and let a,b ∈ D . If a is relatively prime to n , then the congruence ax ≡ b (mod n ) has a solution x ; moreover, any integer x is a solution if and only if x ≡ x (mod n ) . Theorem 6.9 Let n be a non-zero polynomial and let a,b ∈ D . Let d = gcd( a,n ) . If d | b , then the congruence ax ≡ b (mod n ) has a solution x , and any integer x is also a solution if and only if x ≡ x (mod n/d ) . If d- b , then the congruence ax ≡ b (mod n ) has no solution x . 46 Theorem 6.10 (Chinese Remainder Theorem) Let k > , and let a 1 ,...,a k ∈ D , and let n 1 ,...,n k be non-zero polynomials such that gcd( n i ,n j ) = 1 for all 1 ≤ i < j ≤ k . Then there exists a polynomial x such that x ≡ a i (mod n i ) ( i = 1 ,...,k ) . Moreover, any other polynomial x is also a solution of these congruences if and only if x ≡ x (mod n ) , where n := Q k i =1 n i . If we set R = D/nD and R i = D/n i D for 1 ≤ i ≤ k , then in ring-theoretic language, the Chinese Remainder Theorem says the homomorphism from the ring D to the ring R 1 × ··· × R 1 that sends x ∈ D to ([ x mod n 1 ] ,..., [ x mod n k ]) is a surjective homomorphism with kernel nD , and hence R ∼ = R 1 × ··· × R k ....
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