MA352s10midterm_2_solutions

MA352s10midterm_2_solutions - College of the Holy Cross...

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College of the Holy Cross, Spring 2010 Math 352, Midterm 2 Monday, April 19 You may use any results from class or from the text, except for homework exercises unless explicitly stated otherwise. To get full credit for your answers, you must fully explain your reasoning. 1. Give an example of a degree 4 polynomial in Z 5 [ x ] that has no multiple roots in any extension of Z 5 . Give an example of a degree 4 polynomial in Z 5 [ x ] that has no roots in Z 5 but has multiple roots in some extension of Z 5 . Consider f ( x ) = 5 x 4 + x . We have f 0 ( x ) = 1, and so clearly f ( x ) and f 0 ( x ) can have no common factor of positive degree (since f 0 ( x ) has no factor of positive degree at all). Thus by Theorem 20.5, f ( x ) has no multiple roots in any extension of Z 5 . Note that there is no requirement in Theorem 20.5 that the polynomial be irreducible. Now consider f ( x ) = ( x 2 + 2) 2 . By checking all the elements of Z 5 , one sees that f ( x ) has no roots in Z 5 . However, f 0 ( x ) = 4 x ( x 2 + 2), and so f ( x ) and f 0 ( x ) have the common factor of x 2 + 2. Thus by Theorem 20.5, f ( x ) has multiple roots in some extension of Z 5 .
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2. Let α be a root of the polynomial f ( x ) = 2 x 4 + 6 x - 3 Q [ x ]. (a) Find a basis for Q ( α ) as a vector space over Q (b) Write α - 1 as a linear combination of elements of your basis from part (a). (c) Consider the set S = { c 1 α + c 0 | c 0 , c 1 Q } . Prove or disprove that S is a subspace of Q ( α ). (a) First note that by the Eisenstein criterion, f ( x ) is irreducible over Q . By Theorem 20.3, a basis for Q ( α ) as a vector space over Q is { 1 , α, α 2 , α 3 } , since the degree of f ( x ) is 4.
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