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Unformatted text preview: for if c = p 2 e 1 1 · · · p 2 e n n , then u = p e 1 1 · · · p e n n is an integer, and u 2 = c . Now consider the prime factorizations a = p e 1 1 · · · p e n n and b = q f 1 1 · · · q f m m . Since a and b are relatively prime, the sets of prime divisors occurring in their prime factorizations are disjoint; that is, the p i 6 = q j for all i, j . Hence, the prime factorization of ab is ab = p e 1 1 · · · p e n n q f 1 1 · · · q f m m . If ab is a square, then all the exponents e i , f j must be even; that is, a and b are squares....
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This note was uploaded on 11/15/2010 for the course MATH 417 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Staff
 Algebra, Integers

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