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4108_final - Math 4108 FInal Exam Spring 2010 May 5 2010 1...

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Math 4108 FInal Exam, Spring 2010 May 5, 2010 1. Define the following terms a. Euclidean Domain. b. Constructible number. c. Splitting field. d. Jordan-Holder Theorem. e. Associative algebra. 2. Show that in a principal ideal domain, any nested chain of ideals I 1 I 2 I 3 · · · must necessarily satisfy I k = I k +1 = · · · , for some k 1. 3. Suppose that K and L are finite extensions of a field F . Let M be a finite extension of F such that K, L M , and such that [ M : F ] < [ K : F ] · [ L : F ] . Prove that F is properly contained in K L . 4. Let K be the splitting field over Q of the algebraic number α = radicalBig 2 + 3 . Compute the Galois group of K/ Q – i.e. describe Gal( K/ Q ). 1
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5. Suppose that f ( x ) is a particular polynomial of degree 5 whose Galois group (over Q ) has order exceeding 30. Prove that f ( x ) is not solvable in the radicals. 6. Determine the number of degree 11 monic polynomials in F 3 [ x ] that irreducible (write down the exact number).
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