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Unformatted text preview: Homework 1 for MATH 104 Brief Solutions Problem 1 (a) Let F be an ordered field. Let x F . Show that if x > 0 then- x < , if x , 0 then x 2 > . Solution. If 0 < x , then by axiom (OF1), 0 + (- x ) < x + (- x ), hence- x < 0. For the second assertion, we first show that (- a ) 2 = a 2 . Note that = a = a ((- a ) + a ) = a (- a ) + aa , hence a (- a ) =- aa . We use this to infer (- a )(- a ) + (- aa ) = (- a )(- a ) + (- a ) a = (- a )((- a ) + a ) = (- a )0 = , and therefore (- a )(- a ) = aa . Now, if a > 0, then aa > a = 0 by (OF2). If a < 0, then an argument analogues to the first part show that- a > 0. Then 0 < (- a )(- a ) = a 2 . (b) On the set R 2 define the following operations of addition and multiplication: ( x 1 , x 2 ) + ( y 1 , y 2 ) = ( x 1 + y 1 , x 2 + y 2 ) ( x 1 , x 2 )( y 1 , y 2 ) = ( x 1 y 1- x 2 y 2 , x 1 y 2 + x 2 y 1 ) This turns R 2 into a field. (Note: this is just the field of complex numbers, if one identifies a complex number with an element of R 2 , its real and imaginary part.) Show that no order can be defined on this field that turns it into an ordered field....
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- Fall '08