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ProblemSet1 - Problem Set 1 1 Prove case 2 of the Division...

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Problem Set 1 1. Prove case 2 of the Division Algorithm 2. (1.51*) If ζ is a root of unity, prove that there is a positive integer d with ζ d = 1 such that if ζ k = 1 then d | k 3. (1.56*) Let a and b be integers and let sa + tb = 1 for s, t Z . Prove that a and b are relatively prime. 4. (1.57*) If d = ( a, b ) prove that a/d and b/d are relatively prime. 5. (1.60) If a and b are relatively prime and if each divides an integer n , prove that their product ab also divides n . 6. (1.73) Definition If p is prime, define the p -adic norm of a rational number a as follows: if a 6 = 0, then a = ± p e p e 1 1 · · · p e n n , where p , p 1 , . . . , p n are distinct primes, and we set || a || p = p - e ; if a = 0 set || a || p = 0. Define the p -adic metric by δ p ( a, b ) = || a - b || p . (a) For all rationals a and b , prove that || ab || p = || a || p || b || p and || a + b || p max {|| a || p , || b || p } (b) For all rationals a and b , prove δ p ( a, b ) 0 and δ p ( a, b ) = 0 if and only if a = b . (c) For all rationals a and b prove that δ p ( a, b ) = δ p ( b, a ) (d) For all rationals a ,
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