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**Unformatted text preview: **Qualifying Exam in Linear Algebra and Field Theory August, 2006 Answer as many questions as you can, but be sure to answer sufficiently many questions to comprise 120 points, with at least 30 points from each section. Detailed, completely correct answers to a smaller number of questions will obtain more credit than partial answers to a larger number of questions. This is especially important for earning a Ph.D. level pass. Write your answer to each problem on a separate sheet, placing the problem number in the upper right corner. Be sure to justify all your answers. Do not cite any result which reduces a proof to a triviality. Field Theory 1. [5,5,7,5] Let p be a prime. a. In terms of p , how many monic irreducible degree-3 polynomials in F p [ X ] are there? Prove that your answer is correct. b. Show for any prime p , there is always a field of cardinality p 3 . c. Give explicitly the construction, including addition and multiplication tables, of a field with 8 elements....

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