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Algebra2005Aug - Algebra Preliminary Examination Print name...

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Unformatted text preview: Algebra Preliminary Examination, August 22, 2005 Print name: Score: Show your work, provide all necessary proofs and counterexam- ples. There are 10 problems on 20 pages worth the total of 100 points. Check that you have a complete exam. 1. (a) (5 points) How many elements of order 6 are there in the symmetric group S7? to 1. (continued) (b) (5 points) How many conjugacy classes in S7 consist of elements of order 6? 2. (10 points) Show that a group of order 48 cannot be simple. (continued) 5 3. Let G' be a finite group with subgroups H,K S G. Consider the restriction to K of the left action of G on the left cosets of H in G. (a) (4 points) Show that the stabilizer in K of the coset H : 1H is HflK. (b) (3 points) Show that [K : H H K] S [G : H]. 3. (continued) (C) Conclude [G : HflK] g [G : HMO : K]. 4. Let A be a real, symmetric m X m matrix. (a) (5 points) Show that the eigenvalues of A are real. 4. (continued) (b) (5 points) Show that eigenvectors corresponding to distinct eigenval— ues are orthogonal. 9 5. Let C[0.7r] be the real vector space of continuous real—valued functions defined on the closed interval [0,7r], and let V be the subspace of C[0.7r] spanned by the linearly independent functions 1, cos t, sin t, cos2 t, and sin 2t. For all f,g E V consider the expression B(f,g) = fOW(t+1)f(t)g(t)dt. (a) (2 points) Prove that B(f,g) is a bilinear form on V; first define a bilinear form. (b) (2 points) Give the definition of a symmetric bilinear form. Is B(f}g) symmetric? 10 5. (continued) (c) (3 points) Give the definition of a positive definite real quadratic form and determine whether the quadratic form associated to B(f,g) is positive definite. (d) (3 points) Is there a basis 61, . . . ,em for V, for some m > 0, with respect to which the m x m identity matrix [m is the matrix of B(f,g)? ll 6. (10 points) Find all possible Jordan normal forms of a complex m X m matrix A with the Characteristic polynomial (I2 + 3)2(CIZ + 5)4 if the matrix A + 51m is of rank 7. No proof is needed 12 6. (continued) 13 7 (a) (7 points) Prove that the kernel of the homomorphism (15 : C[z,y] —+ (CM of polynomial rings given by (Mar) 2 t2 and My) = t3 is the principal ideal generated by the polynomial y2 — 1‘3. 14 7. (continued) (b) (3 points) Determine the image of (p' explicitly. 15 8. (a) (2 points) Give the definition of an integral domain. ) Give the definition of the characteristic of a nonntriviai (b) (2 points commutative ring. 16 8. (continued) (C) (3 points) Is there an integral domain of Characteristic 6? Explain. (d) (3 points) Is there an integral domain with 12 elements? Explain. 17 9. Determine the irreducible polynomial for ,3 = J? + W over each of the following fields. (a) (3 PointS) @(fil (b) (3 points) Q(\/l4). 18 9. (continued) (C) (4 points) Q. 19 27m 10. Let g 2 e2“. (a) (5 points) Prove that K = (@(C) is a splitting field for the polynomial $5 — 1 over Q and determine the degree [K : Q]. Use the fact that for a prime p, the cyclotomic polynomial 29"} + app—2 + - - . + a: + 1 is irreducible over (1). 2O 10. (continued) (b) (5 points) Determine the Galois group C(K/Q) explicitly and up to isomorphism. ...
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