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### Algebra2010aug

Course: MATH 699, Spring 2011
School: University of North...
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D. Ph. Algebra Preliminary Exam August 23, 2010 Show the work you do to obtain an answer. Give reasons for your answers. There are 10 questions. Do all parts of all questions. Each question is worth 10 points. Q is the eld of rational numbers. C is the eld of complex numbers. Assume all rings have identity not equal to 0. 1. Prove that there is no simple group of order 42. 2. Let G, H , and K be groups with |G| =...

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D. Ph. Algebra Preliminary Exam August 23, 2010 Show the work you do to obtain an answer. Give reasons for your answers. There are 10 questions. Do all parts of all questions. Each question is worth 10 points. Q is the eld of rational numbers. C is the eld of complex numbers. Assume all rings have identity not equal to 0. 1. Prove that there is no simple group of order 42. 2. Let G, H , and K be groups with |G| = 35, |H | = 60, and |K | = 42. Assume there exist group homomorphisms : G H and : G K with ker = G and ker = G. Prove that ker ker consists of one element. 3. Let T : V W be a surjective linear transformation of vector spaces. Let W1 and W2 be subspaces of W such that W = W1 + W2 . Prove that V = T 1 (W1 ) + T 1 (W2 ). 4. Let G be a group of order 77 acting on a set X with 20 elements. Prove that the action has at least 2 xed points. 5. Let V be a nite dimensional vector space over the complex numbers. Let , be a Hermitian form on V . Let W be a subspace of V and assume that the restriction of , to W is nondegenerate. Prove that V is the direct sum V = W W , where W is the orthogonal complement of W computed with respect to , . 6. An ideal I in a commutative ring R is called primary whenever for all a, b R, if ab I , then either a I bn or I for some integer n 1. Let R be a UFD and r an irreducible element of R. For any xed integer m 1, prove that the ideal I = (rm ) is primary. Be sure to justify the use of UFD carefully. 7. Let R be a commutative ring and M a Noetherian R-module. Let f : M M be a surjective R-module homomorphism. Prove that f is an isomorphism. Hint: Consider the kernels of the compositions f n = f f ... f for n = 1, 2, 3, ... . 1 8. Let A be a matrix over C whose only eigenvalues over C are = 7, and = 3 and suppose that dim ker(A 7I ) = 2 dim ker(A 7I )2 = 3 dim ker(A 7I )3 = 3 dim ker(A 3I ) = 2 dim ker(A 3I )2 = 2 (a) (4 points) Find the Jordan form of the matrix A. (Just the Jordan matrix J , not the basis.) (b) (2 points) Find the minimal polynomial of A. (c) (4 points) Let F = C, V = F n where A is an n n matrix and make V into an F [T ]-module by setting T v = Av and extending linearly. Write V as a direct sum r F [T ] V= mi (T ) i=1 with m1 |m2 |...|mr . 9. Let K be the splitting eld for x7 11x + 11 over Q. (a) (8 points) Prove that there exist at least 7 automorphisms in Aut(K/Q). (That is,| Aut(K/Q)| 7.) (b) (2 points) Can there be exactly 10 automorphisms in Aut(K/Q)? 10. Find the Galois group of the splitting eld of x3 41 over Q. 2
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University of North Carolina School of the Arts - MATH - 699
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University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
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University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
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University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
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University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
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University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
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University of North Carolina School of the Arts - MATH - 680
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University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680