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m322s2

# m322s2 - B U Department of Mathematics Spring 2009 Math 322...

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B U Department of Mathematics Spring 2009 Math 322 - Algebra II, Second Midterm Exam, 21/4/2008, 17:00-19:00 Full Name : Student ID : over 40 Below, Q ( S ) denotes the smallest subfield of C containing Q and S C . 1. [10pts] Prove that Q ( 2 , 3) = Q ( 2 + 3) . Generalize your result (not an extensive generalization, just one step ahead).

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2. Fix p ( x ) = x 3 + 2 x 2 + 4 x + 2 in Z 5 [ x ] . (a) [2pts] Show that p ( x ) is irreducible over Z 5 . Use as little technology as possible. (b) [1pt] Let E be a field extension of Z 5 and α E be a root of p ( x ) . Why is it true that Z 5 ( α ) is a field? In your answer, if you use unnecessary argumentation, you do not get any point. (c) [3pts] With α as above, let x = a 0 + a 1 α + a 2 α 2 and y = b 0 + b 1 α + b

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m322s2 - B U Department of Mathematics Spring 2009 Math 322...

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