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Unformatted text preview: Math508, Fall 2010 Jerry L. Kazdan Problem Set 2 DUE: Thurs. Sept. 23, 2010. Late papers will be accepted until 1:00 PM Friday . 1. Let F be a field, such as the reals or the integers mod 7 and x , y ∈ F . Here- x means the additive inverse of x . a) If xy = 0, then either x = 0 or y = 0 (or both are 0). b) Show that (- 1 ) x =- x . [First I needed to prove that 0 x = 0.] c) Show that (- x )(- y ) = xy . 2. Let F be an ordered field, such as the reals or rationals. a) If x < y , show that x < x + y 2 < y . b) For each x ∈ F , if x 6 = 0, then x 2 > 0. c) If x 2 + y 2 = 0, then x = y = 0. 3. Show that the field of complex numbers cannot be made into an ordered field. In other words, there is no possible definition z ≺ w of order that has the properties of an order relation. [HINT: show that in an ordered field, the equation z 2 + 1 = 0 has no solution.] 4. Let f ( x , y ) : = ( x 2 + y 2 ) 2 + y 4 + 2 xy- x + 7 y . Note that f ( , ) = 0....
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This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.
- Fall '10