f09_mthsc852_midterm2

# f09_mthsc852_midterm2 - MTHSC 852, Midterm 2 Fall 2009...

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MTHSC 852, Midterm 2 Fall 2009 Instructor: Dr. Matthew Macauley Monday, November 10, 2009 Name Instructions Exam time is 120 minutes You may not use notes or books. PLEASE WRITE NEATLY AND COMPLETELY! Work out what you want to say on scratch paper before you write it out on the test. If you have any doubt about what results you may assume, then ask . Question Points Earned Maximum Points 1 25 2 15 3 20 4 20 5 20 Total 100

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1. (a) For each group G below, give an example of a polynomial f ( x ) F [ x ] that has G as its Galois group (no proof needed). Be sure to specify the ﬁeld F . (i) D 4 (ii) S 3 (iii) Z 4 (iv) Z 2 × Z 2 × Z 2 (v) Z 852 (b) Let x be transcendental over Z 5 . Describe the Galois group of Z 5 ( x ) over Z 5 .
(c) Carefully deﬁne what it means for a polynomial to be solvable by radicals. (d) Give an example of a polynomial in Q [ x ] that is not solvable by radicals. You don’t need to write it out explicitly; but carefully describe what properties it should have,

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## This note was uploaded on 03/11/2012 for the course MTHSC 851 taught by Professor Staff during the Fall '08 term at Clemson.

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f09_mthsc852_midterm2 - MTHSC 852, Midterm 2 Fall 2009...

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