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781quiz2

781quiz2 - 18.781 ANALYTIC NUMBER THEORY QUIZ Friday Nov 30...

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18.781 ANALYTIC NUMBER THEORY QUIZ Friday, Nov. 30, 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 4 questions. You have 50 minutes to do all the problems. Question Score Maximum 1 6 2 6 3 6 4(BONUS) 3 Total 18

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Question 1 of 4, Page 2 of 5 Solutions 1. Give an explicit definition of the characters χ q : ( Z /q Z ) × -→ C × , where q is prime. That is, provide a careful definition of such a function χ q and then prove that your definition satisfies the necessary property of a character. Solution: Everything on this quiz can be found someplace in the notes posted online, so I will be relatively brief on these solutions. The character χ q to the modulus q is defined by χ q ( n ) = ζ ν ( n ) q - 1 where ζ q - 1 is a ( q - 1)st root of unity, and ν ( n
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781quiz2 - 18.781 ANALYTIC NUMBER THEORY QUIZ Friday Nov 30...

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