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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.
- Spring '10