m322s2

m322s2 - B U Department of Mathematics Spring 2009 Math 322...

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Unformatted text preview: B U Department of Mathematics Spring 2009 Math 322 - Algebra II, Second Midterm Exam, 21/4/2008, 17:00-19:00 Full Name Student ID : : over 40 Below, Q(S ) denotes the smallest subfield of C containing Q and S ⊂ C. √√ √ √ 1. [10pts] Prove that Q( 2, 3) ∼ Q( 2 + 3). Generalize your result (not an extensive generalization, just = one step ahead). 2. Fix p(x) = x3 + 2x2 + 4x + 2 in Z5 [x]. (a) [2pts] Show that p(x) is irreducible over Z5 . Use as little technology as possible. (b) [1pt] Let E be a field extension of Z5 and α ∈ E be a root of p(x). Why is it true that Z5 (α) is a field? In your answer, if you use unnecessary argumentation, you do not get any point. (c) [3pts] With α as above, let x = a0 + a1 α + a2 α2 and y = b0 + b1 α + b2 α2 be two elements in Z5 (α). Find x · y ∈ Z5 (α). (d) [2pts] Show that f = x3 + 3x + 1 and p(x) above are relatively prime, i.e. a greatest common divisor of them is 1. (e) [4pts] Use part (d) and Euclidean algorithm to find the inverse of f (α) in Z5 [α]. 3. (a) [4pts] Show √ 3 π is algebraic over Q(π ). (b) [6pts] Find the minimal polynomial of √ 3 π over Q(π )? 4. [10pts] Construct a regular pentagon using ruler and compass. Go step by step in your drawing. Tell when you use the compass or the ruler. Hint: ...
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This note was uploaded on 05/04/2011 for the course MATH 321 taught by Professor Talinbudak during the Spring '11 term at BU.

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m322s2 - B U Department of Mathematics Spring 2009 Math 322...

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