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Qualifying Exam in Algebra: January 2005, Exams of Algebra

Acharya Nagarjuna UniversityAlgebra

A qualifying exam in algebra, consisting of 18 problems in the areas of linear algebra, group theory, ring theory, and field theory. Students are required to turn in 10 problems, each on a separate sheet, and show work. Problems cover topics such as the relationship between the dimensions of vector spaces and linear transformations, the diagonalizability of finite order matrices, finding jordan canonical forms, properties of symmetric groups, and ideal theory in commutative rings.

Typology: Exams

2012/2013

Uploaded on 02/23/2013

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QUALIFYING EXAM IN ALGEBRA
January 2005
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 2005

  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. Let U , V , and W be vector spaces over a field F and let S : U → V and T : V → W be linear transformations such that T ◦ S = 0 , the zero map. Show that

dim(W/Im T ) − dim(ker T /Im S) + dim ker S = dim W − dim V + dim U.

  1. Let G = GLn(C) be the multiplicative group of invertible n × n matrices with complex entries and let g be an element of G of finite order. Show that g is diagonalizable.
  2. Let A be a 5 × 5 matrix with complex entries such that A^3 = 0. Find all possible Jordan Canonical Forms for A.

II. Group Theory

  1. Prove that the symmetric group Sn is a maximal subgroup of Sn+1. [Hint: Show that if g ∈ Sn+1 − Sn, then Sn+1 = Sn ∪ SngSn.]
  2. Let G be a finite group and let M be a maximal subgroup of G. Show that if M is a normal subgroup of G, then |G : M | is prime.
  3. Let p be a prime and let G be a non-abelian group of order p^3. (a) Show that the center Z(G) of G and the commutator subgroup of G are equal and of order p. (b) Show that G/Z(G) ∼= Zp × Zp.
  4. Let G be a group acting on the set S and let H be a subgroup of G acting transitively on S. Show that if t ∈ S, then G = GtH, where Gt is the stabilizer of t in G.
  5. Show that if G is a group of order 392 = 2^3 · 72 , then G has a normal subgroup of order 7 or a normal subgroup of order 49.

IV. Field Theory

  1. Let α be algebraic over the field F with minimal polynomial f (x) ∈ F [x] and let K = F [α]. Show that if σ : F → L is a field monomorphism and β ∈ L is a root of f σ(x) ∈ L[x], then σ has a unique extension ˆσ : K → L satisfying ˆσ(α) = β.
  2. Let K be a simple algebraic extension of a field F. Show that there are only finitely many intermediate fields between F and K.
  3. Let η be a complex primitive 11th root of unity and let K = Q(η), where Q is the field of rational numbers. Show that there is a unique extension F of degree 2 of Q contained in K and find q ∈ Q such that F = Q(√q).
  4. Let α =

5 + 2√5. Show that Q(α) is a cyclic Galois extension of Q of degree 4. Find all fields F with Q ⊆ F ⊆ Q(α). [Hint: Show that f (x) = x^4 − 10 x^2 + 5 is the minimal polynomial of α over Q and that the roots of f are ±α, ±

√ 5 α .]

  1. (a) Define Galois Extension.

(b) Show that every finite extension of a finite field is a Galois extension.