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Algebra Exam August 2011: Linear Algebra, Group Theory, Ring Theory, and Field Theory, Exams of Algebra

Acharya Nagarjuna UniversityAlgebra

Information about a qualifying exam in algebra held in august 2011. The exam consists of 18 problems, but students are required to turn in only 10. The problems are divided into four categories: linear algebra, group theory, ring theory, and field theory. Each problem is worth an equal amount, and students are instructed to write only on one side of each sheet of paper and to put their name on each page.

Typology: Exams

2012/2013

Uploaded on 02/23/2013

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QUALIFYING EXAM IN ALGEBRA
August 2011
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 – 3 problems
IV. Field Theory – 3 problems
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
pf3
pf4
pf5

Partial preview of the text

Download Algebra Exam August 2011: Linear Algebra, Group Theory, Ring Theory, and Field Theory and more Exams Algebra in PDF only on Docsity!

QUALIFYING EXAM IN ALGEBRA

August 2011

  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 – 3 problems IV. Field Theory – 3 problems

  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 be an n × n Jordan block. Prove that if B is an n × n matrix that commutes with A, then B is a polynomial in A.
  2. Let A =

(a) Find the characteristic polynomial of A. (b) Find the minimal polynomial of A. (c) Find the dimensions of all eigenspaces of A. (d) Find the Jordan canonical form of A.

  1. Let A, B be complex 3 × 3 matrices having the same eigenvectors. The minimal polynomial of A is (x − 1)^2 and the characteristic polynomial of B is x^3. Show that the minimal polynomial of B is x^2.

III. Ring Theory

  1. Let R be any ring with identity, and n any positive integer. If Mn(R) denotes the ring of n × n matrices with entries in R, prove that Mn(I) is an ideal of Mn(R) whenever I is an ideal of R, and that every ideal of Mn(R) has this form.
  2. Let R be a commutative ring with 1 such that for every x in R there is an integer n > 1 (depending on x) such that xn^ = x. Show that every prime ideal of R is maximal.
  3. Let R be a commutative ring with 1 and let f (x) ∈ R[x] be nilpotent. Show that the coefficients of f are nilpotent. ( HINT: Show first that if f, g are nilpotent, then so is f ± g. )
  4. Let R ⊆ S be commutative domains with the same identity, and assume that S is an integral extension of R. Let I be a nonzero ideal of S. Prove that I ∩ R is a nonzero ideal of R.
  5. Let F be a field. Prove that the polynomial ring F [x] is a PID, and that F [x, y] is not a PID.

IV. Field Theory

  1. Determine the Galois group of x^4 − 4 over Q and identify all the in- termediate fields between Q and the splitting field of x^4 − 4. Use the Fundamental Theorem.
  2. (a) Prove that if E is a finite dimensional extension of F which is gener- ated over F by a set S of elements a satisfying a^2 ∈ F for all a ∈ S, then |E : F | = a power of 2. (b) Give an example that shows “2” cannot be replaced by “3”.
  3. Let E = F (α) be a simple extension of F and fix β ∈ E − F. Prove that α is algebraic over F (β).
  4. Let K be a Galois extension of the field F with Galois group G. Let g(x) be a monic polynomial in F [x] and suppose g(x) splits over K (that is, K contains a splitting field for g(x) over F ) and let ∆ ⊆ K be the set of roots of g(x). Prove that g(x) is a power of an irreducible polynomial over F if and only if G is transitive on ∆.
  5. Let E = Q[α, β] where α^2 ∈ Q, β^2 ∈ Q, and |E : Q| = 4. If γ ∈ E −Q and γ^2 ∈ Q prove that γ is a rational multiple of one of α, β, or αβ. (The Fundamental Theorem may be useful.)