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Qualifying Exam in Algebra - January 2006, Exams of Algebra

Dr. Bhim Rao Ambedkar UniversityAlgebra

A qualifying exam in algebra from january 2006, which covers linear algebra, group theory, ring theory, and field theory. The exam consists of 18 problems, and students are required to solve 10 problems in the specified categories. Each problem is worth the same amount of points. The exam includes problems on finding the characteristic and minimal polynomials, eigenvalues, and eigenspaces of a matrix, proving the existence of square roots of matrices, finding a 2-dimensional subspace, and more. It also covers topics such as cyclic normal subgroups, conjugacy classes, simple groups, solvable groups, integral domains, and fields.

Typology: Exams

2012/2013

Uploaded on 02/23/2013

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QUALIFYING EXAM IN ALGEBRA
January 2006
1. There are 18 problems on the exam. Work and turn in 10 problems, in
the following categories.
I. Linear Algebra — 1 problem
II. Group Theory — 3 problems
III. Ring Theory — 2 problems
IV. Field Theory — 3 problems
Any of the four areas — 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.
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QUALIFYING EXAM IN ALGEBRA

January 2006

  1. There are 18 problems on the exam. Work and turn in 10 problems, in the following categories.

I. Linear Algebra — 1 problem II. Group Theory — 3 problems III. Ring Theory — 2 problems IV. Field Theory — 3 problems Any of the four areas — 1 problem

  1. Turn in only 10 problems. No credit will be given for extra problems. All problems are weighted equally.
  2. Put each problem on a separate sheet of paper, and write only on one side. Put your name on each page.
  3. 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) Find the characteristic polynomial of A. (b) Find the minimal polynomial of A. (c) Find the eigenvalues of A. (d) Find the dimensions of all eigenspaces of A. (e) Find the Jordan canonical form of A.

  1. (a) Prove that a 2 × 2 scalar matrix A over a field F has a square root (i.e., a matrix B satisfying B^2 = A). (b) Prove that a real symmetric matrix having the property that every negative eigenvalue occurs with even multiplicity has a square root. [Hint: Use (a).]
  2. Let A, B, and C be subspaces of the nonzero vector space V satisfying

V = A ⊕ B = B ⊕ C = A ⊕ C. Show that there exists a 2-dimensional subspace W ⊆ V such that each of W ∩ A, W ∩ B, and W ∩ C has dimension 1.

III. Ring Theory

In the following problems, all rings are nonzero rings with 1 and all modules are unital.

  1. Let R be an integral domain. Construct the field of fractions F of R by defining the set F and the two binary operations, and show that the two operations are well-defined. Show that F has a multiplicative iden- tity element and that every nonzero element of F has a multiplicative inverse.
  2. Let R be a commutative ring such that not every ideal is a principal ideal. (a) Show that there is an ideal I maximal with respect to the property that I is not a principal ideal. (b) If I is the ideal of part (a), show that R/I is a principal ideal ring.
  3. Let D be an integral domain. (a) For a, b ∈ D define a greatest common divisor of a and b. (b) For x ∈ D denote (x) = {dx | d ∈ D}. Prove that if (a)+(b) = (d), then d is a greatest common divisor of a and b.
  4. Let R be a commutative ring. (a) Prove that (x) is a prime ideal in R[x] if and only if R is an integral domain. (b) Prove that (x) is a maximal ideal in R[x] if and only if R is a field.
  5. Let M be an R-module that is generated by finitely many simple sub- modules. Prove that M is a direct sum of finitely many simple R- modules.

IV. Field Theory

  1. Let f (x) and g(x) be irreducible polynomials in F [x] of degrees m and n, respectively, where (m, n) = 1. Show that if α is a root of f (x) in some field extension of F , then g(x) is irreducible in F (α)[x].
  2. Let K be an algebraic extension of F. Show that the following are equivalent. (i) Each irreducible polynomial in F [x] with one root in K has all its roots in K. (ii) Every F -isomorphism of K into a fixed algebraic closure is an F -automorphism.
  3. Let f (x) = x^4 + 4x^2 + 2 and let K be the splitting field of f over Q. Show that the Galois group of K over Q is cyclic of order 4.
  4. Let (m, n) = 1 and let ηj denote a complex primitive j-th root of unity for any positive integer j. Show that Q(ηmn) = 〈Q(ηm), Q(ηn)〉 and Q(ηm) ∩ Q(ηn) = Q.
  5. Show that every algebraic extension of a finite field is separable.