Adv Alegbra HW Solutions 30

Adv Alegbra HW Solutions 30 - 30 (iv) For all rationals a ,...

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30 (iv) For all rationals a , b , c , prove δ p ( a , b ) δ p ( a , c ) + δ p ( c , b ) . Solution. δ p ( a , b ) δ p ( a , c ) + δ p ( c , b ) because δ p ( a , b ) =k a b k p =k ( a c ) + ( c b ) k p max {k a c k p , k c b k p } k a c k p +k c b k p = δ p ( a , c ) + δ p ( c , b ). (v) If a and b are integers and p n | ( a b ) , then δ p ( a , b ) p n . (Thus, a and b are close if a b is divisible by a large power of p .) Solution. If p n | a b , then a b = p n u , where u is an integer. But k u k p 1 for every integer u , so that δ ( a , b ) =k a b k p =k p n u k p =k p n k p k u k p p n . At this point, one could assign a project involving completions, p -adic integers, and p -adic numbers. 1.74 Let a and b be in Z . Prove that if δ p ( a , b ) p n , then a and b have the same f rst np -adic digits, d 0 ,..., d n 1 . Solution. This follows from the fact that p n | a if and only if the f rst p -adic digits d 0 ,..., d n 1 are all 0. 1.75 Prove that an integer M 0 is the lcm of a 1 , a 2 ,..., a n if and only if it is a common multiple of a 1 , a 2 ,..., a n which divides every other common multiple. Solution. Consider the set I of all the common multiples of a 1 , a 2 ,... , a n . It is easy to check that I satis f es the hypotheses of Corollary 1.37, so that every number m in I is a multiple of
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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