Preliminary Exam - Spring 1992

Problem 1  

1. Prove that every finitely generated subgroup of $\mathbb{Q}\,^{}$, the additive group of rational numbers, is cyclic.

2. Does the same conclusion hold for finitely generated subgroups of $\mathbb{Q}\,^{}$/$\mathbb{Z}^{}$, where $\mathbb{Z}^{}$ is the group of integers?

Problem 2   Find a square root of the matrix

$$\left( \begin{array}{ccc} 1 & 3 & -3 \\ 0 & 4 & 5 \\ 0 & 0 & 9 \end{array} \right).$$

How many square roots does this matrix have?

Problem 3   Let $f$ be an analytic function in the connected open subset $G$ of the complex plane. Assume that for each point $z$ in $G$, there is a positive integer $n$ such that the $n^{th}$ derivative of $f$ vanishes at $z$. Prove that $f$ is a polynomial.

Problem 4   Show that every infinite closed subset of $\mbox{$\mathbb{R}^{n}$}$ is the closure of a countable set.

Problem 5   Let $f$ be a differentiable function from $\mathbb{R}^{n}$ to $\mathbb{R}^{n}$. Assume that there is a differentiable function $g$ from $\mathbb{R}^{n}$ to $\mathbb{R}^{}$ having no critical points such that $g\circ f$ vanishes identically. Prove that the Jacobian determinant of $f$ vanishes identically.

Problem 6   Let $p$ be a prime integer, $p\equiv 3 \pmod{4}$, and let $\mbox{\bf {F}}_p=\mbox{$\mathbb{Z}^{}$}/p\mbox{$\mathbb{Z}^{}$}$. If $x^4+1$ factors into a product $g(x)h(x)$ of two quadratic polynomials in $\mbox{\bf {F}}_p[x]$, prove that $g(x)$ and $h(x)$ are both irreducible over $\mbox{\bf {F}}_p$.

Problem 7   Let $a_1,a_2,\ldots,a_{10}$ be integers with $1\leq a_i \leq 25$, for $1\leq i \leq 10$. Prove that there exist integers $n_1,n_2, \ldots,n_{10}$, not all zero, such that

$$\prod_{i=1}^{10}a_i^{n_i}=1.$$

Problem 8   Evaluate the integral

$$I = \int_{-\infty}^{\infty}\frac{\sin^3x}{x^3}\,dx\, .$$

Problem 9   Let $p$ be a nonconstant polynomial with real coefficients and only real roots. Prove that for each real number $r$, the polynomial $p-rp'$ has only real roots.

Problem 10   Let $A$ denote the matrix

$$\left( \begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right).$$

For which positive integers $n$ is there a complex 4$\times$4 matrix $X$ such that $X^n=A$?

Problem 11   Find a Laurent series that converges in the annulus
$1<\vert z\vert<2$ to a branch of the function $${\log\left(\frac{z(2-z)}{1-z}\right)}$$.

Problem 12   Let $A$ be a real symmetric $n\times n$ matrix with nonnegative entries. Prove that $A$ has an eigenvector with nonnegative entries.

Problem 13   Let $f$ be a one-to-one $C^1$ map of $\mathbb{R}^{3}$ into $\mathbb{R}^{3}$, and let $J$ denote its Jacobian determinant. Prove that if $x_0$ is any point of $\mathbb{R}^{3}$ and $Q_r(x_0)$ denotes the cube with center $x_0$, side length $r$, and edges parallel to the coordinate axes, then

$$\vert J(x_0)\vert = \lim_{r\to 0} r^{-3}\mathrm{vol}\left(f(Q... ...o x_0}\frac{\Vert f(x)-f(x_0)\Vert^3}{\Vert x-x_0\Vert^3}\cdot$$

Here, $\Vert\cdot\Vert$ is the Euclidean norm in $\mathbb{R}^{3}$.

Problem 14  

1. Prove that $\alpha = \sqrt{5}+\sqrt{7}$ is algebraic over $\mathbb{Q}\,^{}$, by explicitly finding a polynomial $f(x)$ in $\mbox{$\mathbb{Q}\,^{}$}[x]$ of degree $4$ having $\alpha$ as a root.

2. Prove that $f(x)$ is irreducible over $\mathbb{Q}\,^{}$.

Problem 15   Let $S_{999}$ denote the group of all permutations of
$\{1,\ldots,999\}$, and let $G\subset S_{999}$ be an abelian subgroup of order $1111$. Prove that there exists $i\in\{1,\ldots,999\}$ such that for all $\sigma\in G$, one has $\sigma(i)=i$.

Problem 16   Let $x_0=1$ and

$$x_{n+1} = \frac{3+2x_{n}}{3+x_{n}}\,, \qquad n \geq 0 \,.$$

Prove that $${x_{\infty} = \lim_{n\to\infty}x_n}$$ exists, and find its value.

Problem 17   For which positive numbers $a$ and $b$, with $a>1$, does the equation $\log_a x = x^b$ have a positive solution for $x$?

Problem 18   Let the function $f$ be analytic in the entire complex plane, real valued on the real axis, and of positive imaginary part in the upper half-plane. Prove $f'(x)>0$ for $x$ real.