practice_midterm_2_problems_solns

practice_midterm_2_problems_solns - College of the Holy...

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College of the Holy Cross, Fall 2009 Math 352, Practice Midterm 2 solutions Prof. Jones 1. Find the splitting field of f ( x ) = x 3 + x + 1 Z 2 [ x ] over Z 2 . First note that f ( x ) is irreducible over Z 2 by Theorem 17.1. Now consider Z 2 ( α ), which by Theorem 20.3 is the same as the set { c 2 α 2 + c 1 α + c 0 | c i Z 2 } . If all three roots of f ( x ) are in this set, Z 2 ( α ) is a splitting field for f ( x ) and we are done. Otherwise we must take an extension of Z 2 ( α ) in order to get all the roots of f ( x ). Note that since α 3 + α + 1 = 0, we have ( α 2 ) 3 + α 2 + 1 = ( α 3 ) 2 + α 2 + 1 2 = ( α 3 + α + 1) 2 = 0 , where the middle equality follows since we’re in characteristic 2. Thus α 2 is also a root of f ( x ). Applying the same reasoning to α 2 gives that α 4 is also a root of f ( x ), and α 4 = α ( α 3 ) = α ( - α - 1) = α 2 + α . Thus we have found the three roots of f ( x ) namely α , α 2 , and α 2 + α , and they all lie in Z 2 ( α ). Hence Z 2 ( α ) is a splitting field for f ( x ). 2. (a) The extension Q ( 6 5) is a vector space over Q . Find a basis for it. Note that 6 5 is a root of the irreducible polynomial x 6 - 5 Q [ x ]. By Theorem 20.3, every element of Q ( 6 5) can be uniquely written as c 5 6 5 5 + c 4 6 5 4 + c 3 6 5 3 + c 2 6 5 2 + c 1 6 5 + c 0 , with c i Q . Thus the set { 1 , 6 5 , 6 5 2 , 6 5 3 , 6
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