Adv Alegbra HW Solutions 23

Adv Alegbra HW Solutions 23 - 23 Solution. Define I = {k...

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Unformatted text preview: 23 Solution. Define I = {k in Z : ζ k = 1}. Of course, I is a subset of Z, and it contains positive numbers because ζ is a root of unity. Observe that (i) If k is in I and u is in Z, then uk is in I : If k is in I , then ζ k = 1. Hence, ζ uk = (ζ k )u = 1u = 1, and so uk is in I . (ii) If k , are in I , then k + is in I : If k , are in I , then ζ k = 1 = ζ , so that 1 = ζ k ζ hence, k + is in I . = ζ k+ ; It now follows from Corollary 1.37 that there is a positive number d in I , i.e., ζ d = 1, such that every k in I is a multiple of d ; that is, if ζ k = 1, then d | k . 1.52 Show that every positive integer m can be written as a sum of distinct powers of 2; show, moreover, that there is only one way in which m can so be written. Solution. In base 2, the only digits are 0 and 1. If we neglect the binary digits 0, then every positive integer is uniquely a sum of powers of 2. 1.53 Find the b-adic digits of 1000 for b = 2, 3, 4, 5, and 20. Solution. base 2 : base 3 : base 4 : base 5 : base 20 : 1000 = 1111101000 1000 = 1101001 1000 = 33220 1000 = 13000 1000 = 2θ, where θ is a symbol denoting the new digit “ten.” 1.54 (i) Prove that if n is squarefree (i.e., n > 1 and n is not divisible by √ the square of any prime), then n is irrational. Solution. We rewrite the proof of Proposition 1.14. Suppose, on √ the √ contrary, that n is rational, where n is squarefree; that is, n = a /b. We may assume that a /b is in lowest terms; that is, (a , b) = 1. Squaring, a 2 = nb2 . Let p be a prime divisor of n , so that n = pq . Since n is squarefree, ( p , q ) = 1. By Euclid’s lemma, p | a , so that a = pm , hence p 2 m 2 = a 2 = pqb2 , and pm 2 = qb2 . By Euclid’s lemma, p | b, contradicting (a , b) = 1. √ (ii) Prove that 3 2 is irrational. √ 3 Solution. Assume that 4 = a /b, where (a , b) = 1. Then 4b3 = a 3 , so that a is even; say, a = 2m . Hence 4b3 = 8m 3 ; canceling, b3 = 2m 3 , forcing b to be even. This contradicts (a , b) = 1. ...
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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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