RUTGERS UNIVERSITY

GRADUATE PROGRAM IN MATHEMATICS

Written Qualifying Examination

January 2005, Day 1

This examination will be given in two three-hour sessions, one today and one tomorrow. At each session the examination will have two parts. Answer all three of the questions in Part I (numbered 1–3) and three of the six questions in Part II (numbered 4–9). If you work on more than three questions in Part II, indicate clearly which three should be graded. No additional credit will be given for more than three partial solutions. If no three questions are indicated, then the first three questions attempted in the order in which they appear in the examination book(s), and only those, will be the ones graded.

First Day—Part I: Answer each of the following three questions

1. Prove that the set ℚ of all rational numbers is not a G_{δ}-subset of the real line.

2. Suppose that a is a real number satisfying a > 1. Use the Residue Theorem to compute

∫_{0}^{2π} \frac{1}{a + cos θ} dθ.

3. Let A be an n × n symmetric real matrix with n distinct eigenvalues. Prove that every n × n real matrix commuting with A is symmetric.

First Day—Part II: Answer three of the following questions. If you work on more than three questions, indicate clearly which three should be graded.

4. Suppose U is an open subset of ℂ. If f is a holomorphic function in U, then a holomorphic function g is called a holomorphic logarithm of f if e^{g} = f in U.

a) Prove that if U is simply connected, then a holomorphic function f which is never 0 always has a holomorphic logarithm, g.

b) Suppose that a holomorphic function f has a holomorphic logarithm g in a connected open subset U of ℂ. Describe with explanation all holomorphic logarithms of f in U.

5. Let p be a prime number. Prove that the polynomial

f(x) = x^{p−1} + x^{p−2} + ⋯ + 1

is irreducible over the ring of rational numbers.

6. Let (X, d) be a metric space. Suppose that for any ϵ > 0 there exists a finite set Y_{ϵ} ⊂ X such that for any x ∈ X

min_{y∈Y_{ϵ}} d(x, y) ≤ ϵ.

Prove that the completion of X is compact.

7. Let f : [0, 1] → ℝ be a function of bounded variation. Prove or construct counter-examples to the following statements:

a) If ϕ : ℝ → ℝ is continuous then the composition ϕ(f) is of bounded variation on [0, 1].

b) If ϕ : ℝ → ℝ is of bounded variation on ℝ then the composition ϕ(f) is of bounded variation on [0, 1].

8. Let E_{1}, E_{2}, …, E_{n} be measurable subsets of a segment [a, b]. Suppose that any point x belongs to at least q of those sets. Prove that at least one of the sets E_{1}, E_{2}, …, E_{n} has measure greater or equal to (b − a)\frac{q}{n}.

9. Suppose that the symmetric group Σ_{4} acts transitively on a finite set X having 8 elements. How many different subgroups of Σ_{4} can occur as stabilizers of points of X?

Day 1 Exam End

RUTGERS UNIVERSITY

GRADUATE PROGRAM IN MATHEMATICS

Written Qualifying Examination

January 2005, Day 2

This examination will be given in two three-hour sessions, today’s being the second part. At each session the examination will have two parts. Answer all three of the questions in Part I (numbered 1–3) and three of the six questions in Part II (numbered 4–9). If you work on more than three questions in Part II, indicate clearly which three should be graded. No additional credit will be given for more than three partial solutions. If no three questions are indicated, then the first three questions attempted in the order in which they appear in the examination book(s), and only those, will be the ones graded.

Second Day—Part I: Answer each of the following three questions

1. Let E ⊂ ℝ be a non-measurable set, and A ⊂ ℝ a set of measure zero. Prove that the set E ∩ (ℝ \ A) is not measurable.

2. Let f : ℂ → ℂ be a holomorphic function such that there is a constant R > 0 such that |z| > R implies |f(z)| > R. Show that f(z) is a polynomial.

3. Prove that no group of order 30 is simple.

Second Day—Part II: Answer three of the following questions. If you work on more than three questions, indicate clearly which three should be graded.

4. Suppose that h(x, y) is a harmonic function on ℝ^{2}, that h(x, y) is never zero and that h(−1, 2) = 1766. What is h(x, y)?

5. Let a + bi be a prime in the Gaussian integers ℤ[i] with a, b ≠ 0. Prove that the integer a^{2} + b^{2} is prime in ℤ.

6. Define f : [0, 1] → ℝ as follows: If x ∈ [0, 1] has decimal expansion x = 0.n_{1}n_{2}n_{3} … then

f(x) = max_{i} n_{i}.

Prove that the function f(x) is measurable.

7. Let f ∈ L^{1}(ℝ). Prove that

∫_{ℝ} |f(x + ϵ) − f(x)|dx

goes to 0 as ϵ goes to 0.

8. Consider the two one-dimensional subspaces of ℂ^{2}, L_{1} = {(z_{1}, 0) : z_{1} ∈ ℂ} and L_{2} = {(0, z_{2}) : z_{2} ∈ ℂ}. Show that the complement ℂ^{2} \ (L_{1} ∪ L_{2}) is connected.

9. Let Q denote the first quarter of the unit disk, that is, Q = {z ∈ ℂ : ℜz > 0, ℑz > 0, 0 < |z| < 1}. Find a conformal mapping from Q to the unit disc D_{1} = {z ∈ ℂ : |z| < 1}. Draw pictures of the domain and range of all mappings you may use to describe your answer.

Exam Day 2 End