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MT2 - 20 i and J = h 18 30 i Find a single generator for...

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Math 113 (Bernd Sturmfels), Midterm Exam # 2 Tuesday, August 1, 9:00 a.m.–10:00 a.m. Please start by writing your name and your student ID on the cover of your blue book. This exam is closed book . Do not use any notes, calculators, cell phones etc. Show all your work, and write full sentences if time permits. Each problem is worth 20 points, for a total of 100 points. (1) Classify all groups of order 65 up to isomorphism. (2) Show that the alternating group A 4 is generated by the two simple 3-cycles (123) and (234). (3) The ring of integers Z is a principal ideal domain, so each ideal in Z is generated by a single element. Consider the ideals I
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Unformatted text preview: 20 i and J = h 18 , 30 i . Find a single generator for each of I , J , I + J , I ∩ J , and I · J . (4) Can you find examples of ideals I in commutative rings R with the following properties ? (a) I is principal and maximal, (b) I is principal but not prime, (c) I is maximal but not prime, (d) I is maximal but not principal, (e) I is principal and prime but not maximal. (5) Consider the two polynomials f = x 4 + x 3 + 7 and g = x 2-2. (a) Compute the remainder of f divided by g . (b) Compute the remainder of g divided by the result of part (a). (c) Determine the ideal in Q [ x ] generated by f and g ....
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