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Homework_2 - Homework 2 Math 139 Due Sep 13th Instructor...

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Homework 2 - Math 139 Due Sep 13th Instructor Mauro Maggioni Office 293 Physics Bldg. Office hours Monday 1:30pm-3:30pm. Web page www.math.duke.edu/˜mauro/teaching.html Reading : from Reed’s textbook: Sections 2.1,2.2 Problems : § 1.1: #8, 10, 11 (in these problems assume only that x and y are elements of an ordered field F ; in #11 assume in addition that F is Archimedean. In the hint for #11, use the well-ordering principle to show that m exists) § 1.4: #9, 11 (You don’t need a. to do c. ; try using b. and #11 above) Additional Problems: 1 . Prove that the field C of complex numbers cannot be given the structure of an ordered field. ( Suggestion: Argue by contradiction: suppose a subset P C exists with the required properties; then i P ( - P ), where i is the complex number such that i 2 = - 1. Deduce the contradiction from this.) 2 .. Let F be a field. Prove that if there is an integer n N such that 1 + 1 + · · · + 1 ( n terms) = 0, then there is no subset P F saisfying the axioms of an odered field.
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