Exam3Review - H of even integers is a subgroup of G . (d)...

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Math 302 Abstract Algebra Spring 2009 Exam 3 Information Date: Friday, May 8 Covering: Chapter 4, Section 6 Chapter 15 Chapter 7, Sections 1, 2, 3, 4 Exam 3 Review Problems 1. Factor the polynomial f ( x ) = x 4 + x 2 12 as a product of irreducible polynomials in each of the following polynomial rings. Indicate why each factor is in fact irreducible. (a) Q [ x ] (b) R [ x ] (c) C [ x ] 2. Show that the number r = 4 p 5 + 2 is constructible. 3. Show that a 40 angle is not constructible. Use this to show that a regular nonagon (9-sided polygon) is not constructible. [ Hint: Use the trigonometric identity cos 3 t = 4 cos 3 t 3 cos t . You can assume the result that an angle of t degrees is constructible if and only if cos t is constructible.] 4. Prove that an angle of t degrees is constructible if and only if sin t is a constructible number. 5. Find the symmetry group D 5 for a regular pentagon. 6. Let G = Z , the set of integers, under the operation a b = a + b 2 for all a,b G . (a) Show that G is a group with the operation . (b) Show that G is cyclic with generator a = 3. (c) Show that the set
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Unformatted text preview: H of even integers is a subgroup of G . (d) Let G = Z , the set of integers under ordinary integer addition. Let f : G G be dened by f ( a ) = 2 a for each a G . Show that f is an isomorphism. 7. Let G be a group and let H be a subgroup of G . Let N ( H ) be the subset of all elements in G that commute with all elements of H ; that is, N ( H ) = { g G | ( h H ) ( gh = hg ) } . Show that N ( H ) is a subgroup of G . 8. Let U 18 = { 1 , 5 , 7 , 11 , 13 , 17 } be the group of units in Z 18 . (a) Find the order of each element in U 18 . (b) Show that U 18 is cyclic and nd all the generators of U 18 . 9. Prove Theorem 7.8(3): Let G be a group and suppose a G . If a has nite order n , then a k = e if and only if n | k. 10. Suppose G and H are groups and f : G H is a surjective homomorphism. Prove that if G is abelian, then H is also abelian....
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