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m25-hw2 - b-1-1 = a-1 b 2.2 Is the following statement...

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Math 25: Advanced Calculus Fall 2010 Homework # 2 Due Date: Friday, October 8 Reading: Finish reading Appendix A of the textbook. Read sections 1.1-1.3. Complete the following problems from the textbook: A.8.1, A.8.4, A.9.1, A.9.2, In addition, please complete the following problems: 2.1: Suppose F is a field. Using only the field axioms, prove the following statements. Please justify each computation you make by citing the appropriate axiom from page 5 of the textbook or citing lemmas we proved in class (Be careful, your algebraic intuition can very easily trick you!). (I) If a, b F , then ( - a ) · b = - ( a · b ). (II) If a, b, c F , then a · ( b + c ) = a · b + a · c . (In class (and in the book), the distributivity axiom was ( x + y ) · z = x · z + y · z . Notice that there could potentially be a difference between these two statements). (III) If a, b F are nonzero, then (
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Unformatted text preview: b-1 )-1 = a-1 · b 2.2: Is the following statement true or false? If x , y , and z are irrational numbers, then x + y + z is irrational. ( Hint: You did some homework last week. One of the problems may be useful!) 2.3 Let R ( x ) denote the collection of rational functions : R ( x ) = ± p ( x ) q ( x ) : p ( x ) ,q ( x ) are polynomials, and q ( x ) 6 = 0 ² 1 . Show that R ( x ) is a field. 1 Notice that there is a different between q ( x ) = 0, which is the function whose output is 0, indepen-dently of the input value x ; and a polynomial function like q ( x ) = x 2 + 2 x-3, which has zeros (x=-3,1), but is not identically equal to 0. The latter is perfectly valid: 1 x 2 +2 x +1 ∈ R ( x ), while the former: p ( x ) is nonsense. 1...
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