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18 Pages

### Algebra2008Jan

Course: MATH 699, Spring 2011
School: University of North...
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Word Count: 1082

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Preliminary Algebra Examination, January 10, 2008 Print name: Show your work, give reasons for your answers, provide all necessary proofs and counterexamples. There are 10 problems on 18 pages worth 10 points each for the total of 100 points. Check that you have a complete exam, print your name on each page. Problem 1 Problem 6 Problem 2 Problem 7 Problem 3 Problem 8 Problem 4 Problem 9 Problem 5 Problem...

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Preliminary Algebra Examination, January 10, 2008 Print name: Show your work, give reasons for your answers, provide all necessary proofs and counterexamples. There are 10 problems on 18 pages worth 10 points each for the total of 100 points. Check that you have a complete exam, print your name on each page. Problem 1 Problem 6 Problem 2 Problem 7 Problem 3 Problem 8 Problem 4 Problem 9 Problem 5 Problem 10 Total Do NOT write on this page 1 Print name: 1. Let G, H be cyclic groups generated by elements x, y of nite orders m, n, respectively. (a) (7 points) Determine the necessary and sucient condition on m, n so that sending xi to y i , for all i Z, is a well-dened homomoprhism of groups. 2 Print name: 1. (continued) (b) (3 points) Describe all homomorphisms of the cyclic group of order 6 into the cyclic group of order 24. 3 Print name: 2. Given a subgroup K of a group G, the set S of left cosets of K in G is a left G-set by means of g xK = gxK , for all g, x G. If H is another subgroup of G, then S is a left H -set by restriction. Recall that the set HxK = {y G | y = hxk for some h H, k K } is called a double coset. For any set X , |X | denotes the cardinality of X . (a) (3 points) Prove that the orbit of the element xK of the H -set S is the set of left cosets of K in G contained in the double coset HxK and compute the stabilizer of xK . (b) (2 points) Prove that the double cosets form a partition of G. 4 Print name: 2. (continued) In the rest of the problem, assume |G| < . (c) (3 points) Prove that |HxK | = |K |[H : H xKx1 ] = |H |[K : K x1 Hx]. (d) (2 points) Do all double cosets have the same cardinality? If yes, give a proof; if no, give a counterexample. 5 Print name: 3. (10 points) Prove that if a group has order pe a where p is a prime, 1 a < p, and e 1, then the group has a proper normal subgroup. 6 Print name: 4. Let A be a square matrix over the eld C of complex numbers. (a) (3 points) Prove that the matrix is invertible if and only if all of its eigenvalues are dierent from zero. (b) (3 points) Prove that the matrix is nilpotent if and only if zero is its only eigenvalue. Recall that a square matrix B is called nilpotent if B m = 0 for some positive integer m. 7 Print name: 4. (continued) (c) (4 points) Prove that if A is nilpotent, it is similar to an upper triangular matrix with diagonal entries zero. Recall that matrices X and Y are called similar if X = CY C 1 for some invertible matrix C . 8 Print name: 5. Let A be a real symmetric n n matrix, and let T : Rn Rn be the linear operator on the Euclidean space Rn given by T (X ) = AX , for all column vectors X Rn . (a) (3 points) Prove that every vector in Ker T is orthogonal to every vector in Im T . (b) (2 points) Prove that Rn = Ker T Im T . 9 Print (continued) (c) name: 5. (5 points) Prove that T is an orthogonal projection onto Im T if and only if A, in addition to being symmetric, satises A2 = A. Recall that for any subspace W Rn , the equality Rn = W W says that every vector v Rn can be uniquely written as v = w + w , where w W and w W . The linear operator on Rn sending v to w, for all v , is called the orthogonal projection onto W . 10 Print name: 6. (10 points) Let Z[x] be the ring of polynomials in one variable with coecients in the integers. Let (3, x) = M Z[x] be the ideal generated by 3 and x. Prove that M is a maximal ideal. 11 Print name: 7. Let R be a commutative ring with identity. Let I and J be ideals of R. Recall that IJ equals the ideal generated by {ij | i I, j J }. (a) (4 points) Prove that IJ I J . (b) (3 points) Give an example where IJ = I J . Make the example nontrivial in the sense that neither I nor J equals either 0 or R. 12 Print name: 7. (continued) (c) (3 points) Give an example where IJ = I J . 13 Print name: 8. Let F be a eld and F [x] the ring of polynomials in one variable with coecients in F . (a) (2 points) Show that a module M over F [x] is also in a natural way a vector space over F . (b) (4 points) Assume that F is algebraically closed and that M is a simple module over F [x]. Prove that the dimension of M as a vector space over F is one. 14 Print name: 8. (continued) (c) (4 points) Assume that F is not algebraically closed. Prove that there exists a simple module M over F [x] such that the dimension of M as a vector space over F is greater than one. 15 Print name: 9. Let T be a linear operator on a nite dimensional vector space over the complex numbers. Assume that T has two eigenvalues: 3,4. Assume that the Jordan canonical form of a matrix representing T has the following form. For the eigenvalue 3 there are 2 blocks of size 1, 2 blocks of size 2, and 1 block of size 4. For the eigenvalue 4 there are 1 block of size 1, 3 blocks of size 3, and 1 block of size 5. (a) (4 points) What is the characteristic polynomial of T ? (b) (3 points) What is the minimal polynomial of T ? 16 Print name: 9. (continued) (c) (3 points) What is the nullity of (T 3I )3 ? I is the identity linear transformation. 17 Print name: 10. (10 points) Let F K be an extension of elds of characteristic 0. Let G be the Galois group of K over F . We do not assume that the eld extension F K is a Galois extension. Assume that G is a nite group and that p is a prime number that divides the order of G. Prove that there exists a eld L with F L K satisfying all of the following properties. (a) L K is a Galois eld extension with Galois group isomorphic to Z/pZ. (b) The degree of the eld extension L K is p. (c) There does not exist any eld strictly between L and K . 18
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University of North Carolina School of the Arts - MATH - 699
Ph. D. Algebra Preliminary Exam Tuesday, August 25, 2009Show 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 worth10 points. When a question has two parts (a) an
University of North Carolina School of the Arts - MATH - 699
ALGEBRA PHD PRELIMINARY EXAM, 9 JANUARY 20091. (10 points) Let G be a group of order 132 = 22 3 11. Prove that G is not simple.2. (10 points) Let H and K be normal subgroups of a group G , and assume that G = HK .Prove that there is an isomorphismG /(
University of North Carolina School of the Arts - MATH - 699
Ph. D. Algebra Preliminary Exam August 23, 2010Show 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 10points. Q is the eld of rational numbers. C is the el
University of North Carolina School of the Arts - MATH - 699
NAME:Algebra Ph.D. Preliminary ExamJanuary 12, 20101. Recall that a subgroup H of a group G is called characteristic if (H ) H for every automorphism of G.(a) Prove that characteristic subgroups are always normal.(b) Let P be a p-Sylow subgroup of a
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
Syllabus for Preliminary Exam for 631-632 Algebra I, IILinear algebra: vector spaces, linear transformations, eigenvectors anddiagonalization, Jordan canonical form, bilinear forms and inner product spaces,normal operators.Groups: cosets, quotient gro
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
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
Analysis Preliminary ExamAugust, 20081. Let f : R2 R be given by the formulax2 yx2 + y 2f (x, y ) =0if (x, y ) = (0, 0)if (x, y ) = (0, 0).(a) Show that f is continuous at (0, 0).(b) Prove that the rst order partial derivatives of f at (0, 0) ex
University of North Carolina School of the Arts - MATH - 600
Analysis Preliminary ExamAugust 24, 20091. If F1 and F2 are closed subsets of R1 and dist(F1 , F2 ) = 0 then F1 F2 = . Prove orgive a counterexample.2. Newtons method for nding zeroes of a function f : R1 R1 is based on the recursionformulaf (xn )x
University of North Carolina School of the Arts - MATH - 600
Analysis preliminary examJan. 8, 20091. Let C be the standard Cantor set on the interval [0, 1] and let A = C c be its complement on the real line. Identify the set of all limit points A of A, explaining your answer.2. (a) Provenk=k=1n(n + 1)2(b)
University of North Carolina School of the Arts - MATH - 600
August 2010 Preliminary Exam in Analysis1. Suppose that f : R R is a function such that f (f (x) = x for all x R. Prove thatthere exists an irrational number t such that f (t) is also irrational.2. Find three subsets A, B, C of the real line R such tha
University of North Carolina School of the Arts - MATH - 600
January 2010 Preliminary Exam in Analysis.1. Let X be a connected metric space. Given two points p, q X and a numberprove that there exist an integer n&gt; 0,0 and points a0 , a1 , . . . , an X such that a0 = p,an = q , andd(aj , aj 1 ) &lt;for all j = 1
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 - 601
University of North Carolina School of the Arts - MATH - 601
University of North Carolina School of the Arts - MATH - 601
Qualifying ExaminationJanuary 10, 2008Algebra Part Please do all ve questions. Problem #5 is worth twice as much as each of the othersWe will always assume that rings have an identity element and that modulesare unitary left modules.1. Let I be an
University of North Carolina School of the Arts - MATH - 601
Algebra P art o f Qualifying Examination, A ugust 25, 2009Instructions: Do all questions, justify your answers with the necessary proofs.All rings are associative (not necessarily commutative) with identity a nd all modulesare left unitary modules. We
University of North Carolina School of the Arts - MATH - 601
Q ualifying Exam - January 2009Algebra PartInstructions: C omplete as m any q uestions a s possible. Answers should be justified withthe necessary proofs. All rings are a ssumed to be n oncom mutative unless statedotherwise. All rings have an identity
University of North Carolina School of the Arts - MATH - 601
Algebra Part of Qualifying Examination, August 23, 2010Instructions: Do all questions, justify your answers with the necessary proofs. Allrings are associative (not necessarily commutative) with identity, and all modules are leftunitary modules. We den
University of North Carolina School of the Arts - MATH - 601
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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
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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
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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
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
QUALIFYING EXAM COMPLEX ANALYSISThursday, January 8, 2009Show ALL your work. Write all your solutions in clear, logical steps. Good luck!Your Name:Problem Score Max120220330430Total100Problem 1. Let f = f (z ) be analytic in the unit disk,
University of North Carolina School of the Arts - MATH - 680
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University of North Carolina School of the Arts - MATH - 680
Complex Part1. Show that the function f (z ) = 1/z has no a holomorphic antiderivative on cfw_&lt; 1 &lt; |z | &lt; 2.2. Suppose that f is an entire function and f 2 is a holomorphicpolynomial. Show that f is also a holomorphic polynomial.3. Suppose that a fun
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
Topics for Qualifying Exam in Complex AnalysisIComplex Plane and Elementary Function.a) Complex Numbersb) Polar Representationc) Stereographic Projectiond) The Square and Square Root Functionse) The Exponential Functionf) The Logarithm Functiong)