{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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?
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern