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Unformatted text preview: 18.781 RECIPROCITY QUIZ Monday, Nov. 5, 2007 Name: Numeric Student ID: Instructor’s Name: I agree to abide by the terms of the honor code: Signature: Instructions: Print your name, student ID number and instructor’s name in the space provided. During the test you may not use notes, books or calculators. Read each question carefully and show all your work ; full credit cannot be obtained without sufficient justification for your answer unless explicitly stated otherwise. Underline your final answer to each question. There are 3 questions. You have 50 minutes to do all the problems. Question Score Maximum 1 7 2 7 3 (BONUS) 5 Total 14 Question 1 of 3, Page 2 of 5 Solutions 1. Answer “TRUE” or “FALSE” to the following questions. You don’t need to show your work. All computations are understood to be occurring in the ring R = Z [ ω ]. (a) 13 ≡ 0 (mod 2 + 3 ω ) Solution: N (2 + 3 ω ) = 2 2 2 · 3 + 3 2 = 7. If (2 + 3 ω ) q = 13 for some q ∈ R , then by multiplicativity of the norm map, N (2 + 3 ω )  N (13). Clearly, 7 13 2 , so FALSE. (b) 20 ≡ 41 (mod 1 ω ) Solution: The assertion is equivalent to 1 ω  (41 20) = 21. Now N (1 ω ) = 3, so (1 ω )  3  21, so TRUE.21, so TRUE....
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This note was uploaded on 04/28/2009 for the course MATH 18.781 taught by Professor Brubaker during the Spring '09 term at MIT.
 Spring '09
 BRUBAKER
 Number Theory

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