103b_07s_fe

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

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Unformatted text preview: 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...
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This note was uploaded on 06/18/2009 for the course MATH 103b taught by Professor Staff during the Spring '08 term at UCSD.

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

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