Midterm_3_practice_problem_solutions

Midterm_3_practice_problem_solutions - College of the Holy...

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College of the Holy Cross, Fall 2009 Math 351 – Solutions to the practice problems for exam 3 #1. Suppose that a is algebraic over Q and f ( x ) is a polynomial with coefficients in Q . Show that Q ( f ( a )) is an algebraic extension of Q . Since f ( x ) has coefficients in Q , we have f ( x ) = b n x n + ··· + b 1 x + b 0 , where a i Q . Thus f ( a ) = b n a n + ··· + b 1 a + b 0 , which is in Q ( a ) since Q ( a ) is closed under field operations. Thus Q ( f ( a )) Q ( a ). Since Q ( a ) is algebraic, it is finite (its degree over Q is equal to the degree of the minimal polynomial of a over Q ). Because Q ( f ( a )) Q ( a ), we have [ Q ( f ( a )) : Q ] [ Q ( a ) : Q ], and thus Q ( f ( a )) is finite. Since finite extensions are algebraic (Theorem 21.4), we have that Q ( f ( a )) is algebraic. #2. Let E = Q ( p (3) , p (5)), and assume that [ E : Q ] = 4. Find the elements of Gal( E/ Q ) (recall
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