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Qualifying Examination in Algebra August 2000 The field of real numbers will be denoted by ~, the field of complex numbers by C and the field of rational numbers by Q. The ring of integers is denoted by Z. Note! You must show sufficient work to support your answer. \Vrite. your answers as legibly as you can on the blank sheets of paper provided. Use only on.e side of each sheet; start each problem on a new sheet of paper; and be sure to number your pages. Put your solution to problem 1 first, and then your solution to number 2, etc. H some problem is incorrect, then give a counterexample. 1. Let 4>: Z E9Z E9Z E9Z -+ Z EB Z EB Z be the group homomorphism which is given by multiplication by the matrix . 4 10 IS 16 lvf= 2 2 6 . 2 [ 6 12 24 18] Express the cokernel of 4>as a direct sum of cyclic groups. (Recall that the cokernel eze~ .) of <pis defined to be . Z Image... . ./ . 2. Let R be the quotient ring R = Q[xJ/ (X3 - 3x - 3), let 7t':Q[xJ -+ R be the quotient map and let x = 7t'(x) be the image of x under the quotient map. (a:) Show thatR is a field. (b) Find the dimension of R as a vector space over Q. (c) express 1/ (1 + x) as a polynomial of degree :5 2 in x. 3. Let G be a finite group of odd order. Show that every subgroup of G of index 3 is nprmal. . 4. Let A(t), B(t) E ~[tJ with deg(A(t)) = ~eg(B(t)) = 5. Then show there is a nonzero polynomial p(x, y) E IR[x,yJ so that p(A(t), B(t)) = O. (That is, p(A(t), B(t)) is the zero polynomial in IR[t].) 5. Let A1nxn(C) be the n x n over the complex numbers. For A E i\1nxn(C) let C;\ := {B E 1Vlnxn(C) : AB = BA} be the set of elements of Mnxn(C) that commute with A. . (a) Show CAis a subspace of Nlnxu(C). (b) Show dimCA 2:n. . (c) Give an example of a matrix A where dimCA = n. 6. Let ~r be a finite dimensional vector space and T: V -+ V a linear map. Let ker(T) be the kernel of T and Image(T) the image of T. If Image(T) ~ Image(T2) then show V = Image(T) E9ker(T). 7. Let I be an ideal in the commutative ring R which is maximal among the nonfinitely generated ideals of R. Pro\'e that I is a prime ideal of R. (The hypothesis on I says that if J is an ideal of R with I c J and I =1= J, then J is a finitely generat~d ideal of R.) 1 8. List 4 proper prime ideals of the polynomial ring Clx, y] which contain the ideal (4X2 - 13xy + 3y2). Which of your ideals are maximal ideals? 9. Do there exist proper subgroups G1 and G2 of the group (Q, +), with G1 EBG2 isomorphic to (Q, +)? 10. (True or False. If true, prove it. If false, give a counterexample.) If H is a normal subgroup of the finite group G, then there exists a subgroup of G which is isomorphic to ~. ... ... .. .. ....--.-....