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Linear Algebra - Abstract Algebra - Exam, Exams of Algebra

Punjab Engineering CollegeAlgebra

This is the Exam of Abstract Algebra which includes Linear Algebra, Group Theorym, Jordan Canonical, Finite Dimensional, Integral Domain, Element, Indicated Group, Units, Definition etc. Key important points are: Linear Algebra, Group Theory, Ring Theory, Field Theory, Characteristic Polynomial, Minimal Polynomial, Jordan Canonical Forms, Matrix, Dimensional Vector Space, Even Integer

Typology: Exams

2012/2013

Uploaded on 02/23/2013

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QUALIFYING EXAM IN ALGEBRA
August 2000
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

August 2000

  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.

I. Linear Algebra

  1. A matrix A has characteristic polynomial ∆(x) = (x − 3)^5 and minimal polynomial m(x) = (x − 3)^3. (a) List all possible Jordan canonical forms for A. (b) Determine the Jordan canonical form of the matrix

A =

       

       

which has the given characteristic and minimal polynomials.

  1. Let V be a finite dimensional vector space and T : V → 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 over a field F and let U be a subspace. Show that there is a subspace W of V such that V = U ⊕ W.

III. Ring Theory

  1. Let p be a prime and let R be the ring of all 2 × 2 matrices of the form

  a^ b pb a

 ,

where a, b ∈ Z. Prove that R is isomorphic to Z[√p].

  1. Let R be a commutative ring with identity. Suppose that for every a ∈ R, either a or 1 − a is invertible. Prove that N = {a ∈ R | a is not invertible} is an ideal of R.
  2. Show that if R is a finite commutative ring with identity then every prime ideal of R is a maximal ideal.
  3. Let D be a unique factorization domain such that if p and q are irreducible elements of D, then p and q are associates. Show that if A and B are ideals of D, then either A ⊆ B or B ⊆ A.
  4. Let D be an integral domain and F a subring of D which is a field. Show that if each element of D is algebraic over F , then D is a field.

IV. Field Theory

  1. Let K be a field extension of F of degree n and let f (x) ∈ F [x] be an irreducible polynomial of degree m > 1. Show that if m is relatively prime to n, then f has no root in K.
  2. A field F is called perfect if every element of an algebraic closure of F is separable over F. Let F be a field of characteristic p. Show that the following are equivalent. (i) The field F is perfect. (ii) The map a 7 → ap^ is an automorphism of F.
  3. Let u =

√ 2 + √2, v =

√ 2 − √2, and E = Q(u), where Q is the field of rational numbers. (a) Find the minimal polynomial f (x) of u over Q. (b) Show v ∈ E. Hence conclude that E is a splitting field of f (x) over Q. (c) Show that the Galois group of E over Q is cyclic of order 4.

  1. Let K be a Galois extension of k and let k ⊆ F ⊆ K and k ⊆ L ⊆ K. (a) Show that Gal(K/LF ) = Gal(K/L) ∩ Gal(K/F ). (b) Show that Gal(K/L ∩ F ) = 〈Gal(K/L), Gal(K/F )〉.
  2. Let F be a finite field. Show that the product of all the non-zero elements of F is −1.