f09_mthsc852_finalexam

# f09_mthsc852_finalexam - MTHSC 852, Final Exam Fall 2009...

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Unformatted text preview: MTHSC 852, Final Exam Fall 2009 Instructor: Dr. Matthew Macauley Wednesday, December 9, 2009 Name Instructions • Exam time is 2.5 hours. • 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. There are two pages of scratch paper at the end. • If you have any doubt about what results you may assume, then ask . Question Points Earned Maximum Points 1 20 2 15 3 20 4 15 5 20 6 10 Total 100 Student to your left: Student to your right: 1. (a) Carefully define what it means for an element in a ring R to be (i) prime , and (ii) irreducible . Which one of these implies the other, and what is the strongest condition on R that guarantees that they are equivalent (no proof needed). (b) Suppose p is a prime and f ( x ) = x n + ··· + a 1 x + a ∈ Z [ x ] with p | a i ∀ i and p 2- a . Prove that f is irreducible over Z [ x ]. 2. Let K/F be a Galois extension of degree n ; let p be a prime dividing...
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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_finalexam - MTHSC 852, Final Exam Fall 2009...

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