4108_midterm1 - Midterm 1, Math 4108, Spring 2010 March 1,...

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Unformatted text preview: Midterm 1, Math 4108, Spring 2010 March 1, 2010 1. Define the following terms. a. Algebraic extension. b. Unique Factorization Domain. c. Splitting field. d. Eisenstein’s criterion. e. algebraic number. 2. Compute the degree of the extension F [x]/(x2 − 2) over Q, where F = Q(i), where i2 = −1. Explain your answer. 3. Determine the gcd of the polynomials f (x) = x5 + 3x3 + 2x2 + 2x + 2 and g (x) = x5 + x4 + 3x3 + 4x2 + 4x + 2, where we think of f, g ∈ F7 [x]. 4. Suppose that K is an algebraic extension of a characteristic 0 field F , in which every α ∈ K satisfies some polynomial of degree at most B > 0 (i.e. we have an upper bound on the degree of elements in K ). Prove that K is in fact a finite extension of F . (Hint: Primitive element theorem. If you don’t know what it says, ask me in class [it is was not on the study sheet... and this problem is too nice not to be on an exam].) 5. Suppose that f (x) ∈ F [x] is a degree n irreducible polynomial, and that E is some finite extension of F in which f (x) has a root. Prove that n divides [E : F ]. 1 ...
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This note was uploaded on 10/23/2011 for the course MATH 4108 taught by Professor Staff during the Spring '10 term at Georgia Institute of Technology.

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