p2 - a 1 a 2 a 3 , where a n = 1 + 1 n n . Hint. Consider a...

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #2 (due on Wednesday, September 21). 50 points are divided between 5 problems, 10 points each. #1. Let F be a field. Show that there exist not more that two different solutions solutions of the equation x · x = 1. Is it possible that there is only one solution to this equation? #2. Let F = { 0 , 1 , 2 , 3 , 4 } be a field such that 1 + 1 = 2 , 1 + 2 = 3 , 1 + 3 = 4 , 1 + 4 = 0 . Find the values of xy for all x,y F . Put the results into the table: y \ x 0 1 2 3 4 0 . . . . . 1 . . . . . 2 . . . . . 3 . . . . . 4 . . . . . Remark. You do not need to verify the axioms of a field. #3. Show that any any two rational numbers p 1 < p 2 there exists an irrational number r such that p 1 < r < p 2 . #4. Show that (1 + h ) n 1 + nh for all natural n and real h such that | h | ≤ 1. (This is Problem 7(a) on p. 22.) Using this result, show that
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Unformatted text preview: a 1 a 2 a 3 , where a n = 1 + 1 n n . Hint. Consider a n /a n-1 . #5. Represent z 4 + 4 in the form z 4 + 4 = ( z-z )( z-z 1 )( z-z 2 )( z-z 3 ) with xed complex numbers z k , and also as a product of two quadratic polynomials with real coecients. Hint. Here z k are distinct complex roots of the polynomial z 4 + 4, or equivalently, solutions of the equation z 4 =-4. In general, z n = w = r e i = r (cos + i sin ) with r &gt; has n distinct solutions z k = r 1 /n e i ( +2 k ) /n , where k = 0 , 1 , 2 ,...,n-1 ....
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This document was uploaded on 02/15/2012.

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