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Qualifying Exam in Algebra - January 2013, Exams of Algebra

Punjab Engineering CollegeAlgebra

The qualifying exam in algebra from january 2013, which covers linear algebra, group theory, ring theory, and field theory. The exam has 18 problems, and students are required to solve 10 problems in the specified categories. Each problem is weighted equally, and students must put each problem on a separate sheet of paper and write only on one side. The exam provides an opportunity for students to demonstrate their understanding of various algebraic concepts and techniques.

Typology: Exams

2012/2013

Uploaded on 02/23/2013

mohan_das
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QUALIFYING EXAM IN ALGEBRA
January 2013
1. There are 18 problems on the exam. Work and turn in 10 problems, in the following
categories.
I. Linear Algebra โ€” 1 problem
II. Group Theory โ€” 3 problems
III. Ring Theory โ€” 2 problems
IV. Field Theory โ€” 3 problems
Any of the four areas โ€” 1 problem
2. Turn in only 10 problems. No credit will be given for extra problems. All problems
are weighted equally.
3. Put each problem on a separate sheet of paper, and write only on one side. Put your
name on each page.
4. If you feel there is a misprint or error in the statement of a problem, then interpret it
in such a way that the problem is not trivial.
1
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QUALIFYING EXAM IN ALGEBRA

January 2013

  1. There are 18 problems on the exam. Work and turn in 10 problems, in the following categories. I. Linear Algebra โ€” 1 problem II. Group Theory โ€” 3 problems III. Ring Theory โ€” 2 problems IV. Field Theory โ€” 3 problems Any of the four areas โ€” 1 problem
  2. Turn in only 10 problems. No credit will be given for extra problems. All problems are weighted equally.
  3. Put each problem on a separate sheet of paper, and write only on one side. Put your name on each page.
  4. If you feel there is a misprint or error in the statement of a problem, then interpret it in such a way that the problem is not trivial.

I. Linear Algebra

  1. Let A =

(a) Verify that the characteristic polynomial of A is โˆ†(x) = x(x โˆ’ 1)^2. (b) For each eigenvalue ฮป of A, find a basis for the eigenspace Eฮป. (c) Determine if A is diagonalizable. If so, give matrices P , B such that P โˆ’^1 AP = B and B is diagonal. If not, explain carefully why A is not diagonalizable.

  1. Let V be a finite dimensional vector space and T : V โ†’ V a non-zero linear operator. Show that if ker T = Im T , then dim V is an even integer and the minimal polynomial of T is m(x) = x^2.
  2. Let V be a finite dimensional vector space and let W be a subspace. Show that dim V /W = dim V โˆ’ dim W.

II. Group Theory

  1. Show that if the center of a group G is of index n in G, then every conjugacy class of G has at most n elements.
  2. Let G be an abelian group. Let K = {a โˆˆ G | a^2 = 1} and let H = {x^2 | x โˆˆ G}. Show that G/K โˆผ= H.
  3. Let G be a group of order pnq, where p and q are distinct primes, and assume q โˆค pi^ โˆ’ 1 for 1 6 i 6 n โˆ’ 1. Prove that G is solvable.
  4. Show that a simple group of order 168 must be isomorphic to a subgroup of the alternating group A 8.
  5. Let H and K be subgroups of a group G with K E G. Show that if H and K are solvable, then HK is solvable.