Homework 2 - Math 139Due Sep 13thInstructorMauro MaggioniOffice293 Physics Bldg.Office hoursMonday 1:30pm-3:30pm.Web pagewww.math.duke.edu/˜mauro/teaching.htmlReading: from Reed’s textbook: Sections 2.1,2.2Problems:§1.1: #8, 10, 11 (in these problems assume only thatxandyare elements of an orderedfieldF; in #11 assume in addition thatFis Archimedean. In the hint for #11, use thewell-ordering principle to show thatmexists)§1.4: #9, 11 (You don’t needa.to doc.; try usingb.and #11 above)Additional Problems:1.Prove that the fieldCof complex numbers cannot be given the structure of anordered field. (Suggestion:Argue by contradiction: suppose a subsetP⊆Cexists withthe required properties; theni∈P∪(-P), whereiis the complex number such thati2=-1. Deduce the contradiction from this.)2..LetFbe a field.Prove that if there is an integern∈Nsuch that 1 + 1 +· · ·+1 (nterms) = 0, then there is no subsetP⊆Fsaisfying the axioms of an odered field.
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