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**Unformatted text preview: **Math 442 Final Examination and Qualifying Examination in Linear Algebra and Field Theory December, 2008 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 Do not cite any result that reduces a proof to a triviality. Field Theory 1. [5,5,5,8] Let K be a field, let f ( X ) be a polynomial in K [ X ] and suppose deg( f ) = n . a. Prove that every element of K [ X ] / h f ( X ) i has a unique representation h ( X )+ h f ( X ) i , where deg( h ) < n . b. Explain in detail why f ( X ) must be irreducible in K [ X ] in order for K [ X ] / h f ( X ) i to be a field. c. Suppose that f ( X ) is irreducible in K [ X ]. Prove that K [ X ] / h f ( X ) i = K ( ), where satisfies f ( ) = 0. d. Suppose that f ( X ) and g ( X ) are irreducible polynomials over the field K and that the greatest common divisor gcd(deg( f ) , deg( g )) = 1. Let be a zero of f in some algebraic closure of K . Show that....

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