s2 - Math 5615H: Introduction to Analysis I. Homework #2....

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #2. Problems and Solutions. #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? Solution. If x · x = 1, then ( x - 1)( x + 1) = x 2 - 1 = 0. Then x = 1 or x = - 1. In the case F := { 0 , 1 } with 1 + 1 = 0, we have - 1 = 1. This is the only case when the given equation has exactly one solution. #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. You do not need to verify the axioms of a field. Answer: y \ x 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 1 3 3 0 3 1 4 2 4 0 4 3 2 1 #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 . Solution. We have p 1 < r := p 1 +( p 2 - p 1 ) / 2 < p 2 . If r is rational, then 2 = ( r - p 1 ) / ( p 2 - p
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This document was uploaded on 02/15/2012.

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