prelim1 - u is an element of R satisfying u 2 = 1, show...

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Mathematics 3360 Prelim 1 February 23, 2010 No books, notes, or electronic devices may be used. You may use anything that has been given in class or in the book, as long as you show clearly what you are using. SHOW ALL OF YOUR WORK AND JUSTIFY YOUR ANSWERS. 1 (20 points) . Is [11] 41 a unit in Z 41 ? If so, find its inverse. If not, say why not. 2 (20 points) . Find all solutions of the congruence 9 x 9 (mod 69) . 3 (20 points) . Let a be an integer not divisible by any prime less than 50. Show that a 60 1 (mod 385). 4 (20 points) . (a) List the units in Z 8 and show that they all satisfy u 2 = 1. (b) Let R be a commutative ring with no 0-divisors. If
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Unformatted text preview: u is an element of R satisfying u 2 = 1, show that u = 1. [Note: You may not assume that R = Z m .] (c) Why doesnt (b) contradict (a)? 5 (15 points) . Let R be the set of all complex numbers of the form a + bi , where a and b are rational numbers. (a) Show that R is closed under the usual addition and multiplication operations. (b) Explain why R is a ring. [Hint: This doesnt require a long answer.] 6 (5 points) . Does there exist a homomorphism Z 2 Z ? If so, exhibit one. If not, explain why not....
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