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103b_07s_fe

103b_07s_fe - Math 103B Final Exam(100 points 11 June 2007...

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Math 103B Final Exam (100 points) 11 June 2007 Please put your name and ID number on your blue book. The exam is CLOSED BOOK except for both sides of two sheets of notes. Calculators are NOT allowed. In a multipart problem, you can do later parts without doing earlier ones. You must show your work to receive credit. 1. (10 pts.) If R is a ring, define its center Z ( R ) to be those elements that commute with all elements of R ; that is, Z ( R ) = { z R | zr = rz for all r R } . Prove that Z ( R ) is a commutative subring of R . 2. (12 pts.) In the following, we are thinking or R and Q as subfields of C , the complex numbers. Therefore, express your answers as subfields of C . (a) Find the splitting field of ( x 2 + 1)( x 2 + x + 1) over R . (b) Find the splitting field of ( x 2 + 1)( x 2 + x + 1) over Q . Remember to explain how you got your answers. 3. (10 pts.) Suppose E F are fields, [ E : F ] = n , f ( x ) F [ x ] is irreducible over F and E contains a zero of f ( x ). What are the possible values for the degree of f ( x )? Why?

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