Preliminary Exam - Summer 1984

Problem 1   Show that if a subgroup $H$ of a group $G$ has just one left coset different from itself, then it is a normal subgroup of $G$.

Problem 2   Let $\mathbb{Z}^{}$ be the ring of integers and $\mbox{$\mathbb{Z}^{}$}[x]$ the polynomial ring over $\mathbb{Z}^{}$. Show that

$$x^6+539x^5-511x+847$$

is irreducible in $\mbox{$\mathbb{Z}^{}$}[x]$.

Problem 3   Let $f: \mbox{$\mathbb{R}^{m}$} \to \mbox{$\mathbb{R}^{n}$}, \; n \geq 2$, be a linear transformation of rank $n-1$. Let $f(v) = (f_1(v),f_2(v),\ldots,f_n(v))$ for $v \in\mbox{$\mathbb{R}^{m}$}$. Show that a necessary and sufficient condition for the system of inequalities $f_i(v)>0$, $i=1,\ldots,n$, to have no solution is that there exist real numbers $\lambda_i\geq 0$, not all zero, such that

$$\sum_{i=1}^n\lambda_if_i=0.$$

Problem 4   Let

$$A = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right)$$

be a real matrix with $a, b, c, d > 0$. Show that $A$ has an eigenvector

$$% latex2html id marker 769 \left( \begin{array}{c} x \\ y \end{array} \right)\in \mbox{$\mathbb{R}^{2}$}$$

with $x, y > 0$.

Problem 5  

1. Show that there is a unique analytic branch outside the unit circle of the function $f(z)=\sqrt{z^2+z+1}$ such that $f(t)$ is positive when $t>1$.

2. Using the branch determined in Part 1, calculate the integral
   
   $$\frac{1}{2\pi i}\int_{C_r}\frac{dz}{\sqrt{z^2+z+1}}$$
   
   
   where $C_r$ is the positively oriented circle $\vert z\vert=r$ and $r>1$.

Problem 6   Let $\rho > 0$. Show that for $n$ large enough, all the zeros of

$$f_n(z) = 1 + \frac{1}{z} + \frac{1}{2!z^2} + \cdots + \frac{1}{n!z^n}$$

lie in the circle $\vert z\vert<\rho$.

Problem 7   Let $f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$ be $C^1$ and let

$$\begin{array}{l} u = f(x) \\ v = -y + xf(x). \end{array}$$

If $f'(x_0)\neq 0$, show that this transformation is locally invertible near $(x_0,y_0)$ and the inverse has the form

$$\begin{array}{l} x = g(u) \\ y = -v + ug(u). \end{array}$$

Problem 8   Let $\varphi(s)$ be a $C^2$ function on $[1,2]$ with $\varphi$ and $\varphi'$ vanishing at $s=1,2$. Prove that there is a constant $C>0$ such that for any $\lambda>1$,

$$\left\vert\int_1^2e^{i\lambda x}\varphi(x)\,dx\right\vert \leq \frac{C}{\lambda^2}\cdot$$

Problem 9   Consider the solution curve $\left( x(t),y(t) \right)$ to the equations

$$\begin{eqnarray*} \frac{dx}{dt} & = & 1 + \frac{1}{2}x^2\sin y \\ \frac{dy}{dt} & = & 3 - x^2 \end{eqnarray*}$$

with initial conditions $x(0)=0$ and $y(0)=0$. Prove that the solution must cross the line $x=1$ in the $xy$ plane by the time $t=2$.

Problem 10   Let $C^{1/3}$ be the set of real valued functions f on the closed interval $[0,1]$ such that

1. $f(0)=0$;

2. $\Vert f\Vert$ is finite, where by definition
   
   $$\Vert f\Vert = \sup\left\{\frac{\vert f(x)-f(y)\vert}{\vert x-y\vert^{1/3}} \;\vert\; x\neq y \right\}.$$
   
   

Verify that $\Vert\cdot\Vert$ is a norm for the space $C^{1/3}$, and prove that $C^{1/3}$ is complete with respect to this norm.

Problem 11   Let $S_n$ denote the group of permutations of $n$ letters. Find four different subgroups of $S_4$ isomorphic to $S_3$ and nine isomorphic to $S_2$.

Problem 12   Let $\mbox{\bf {F}}_q$ be a finite field with $q$ elements and let $V$ be an $n$-dimensional vector space over $\mbox{\bf {F}}_q$.

1. Determine the number of elements in $V$.
2. Let $GL_n(\mbox{\bf {F}}_q)$ denote the group of all $n\times n$ nonsingular matrices $A$ over $\mbox{\bf {F}}_q$. Determine the order of $GL_n(\mbox{\bf {F}}_q)$.
3. Let $SL_n(\mbox{\bf {F}}_q)$ denote the subgroup of $GL_n(\mbox{\bf {F}}_q)$ consisting of matrices with determinant $1$. Find the order of $SL_n(\mbox{\bf {F}}_q)$.

Problem 13   Let $A$ be a 2$\times$2 matrix over $\mathbb{C}\,^{}$ which is not a scalar multiple of the identity matrix $I$. Show that any 2$\times$2 matrix $X$ over $\mathbb{C}\,^{}$ commuting with $A$ has the form $X=\alpha I+\beta A$, where $\alpha, \beta \in \mbox{$\mathbb{C}\,^{}$}$.

Problem 14   Suppose $V$ is an $n$-dimensional vector space over the field $\mbox{\bf {F}}$. Let $W\subset V$ be a subspace of dimension $r<n$. Show that

$$W = \bigcap\,\left\{U\;\vert\; U\, is \, an \, (n-1)-dimensional\; subspace\; of\; V\, and\;W \subset U\right\}.$$

Problem 15   Let $\mbox{$\mathbb{Z}^{}$}_3$ be the field of integers $\bmod 3$ and $\mbox{$\mathbb{Z}^{}$}_3[x]$ the corresponding polynomial ring. Decompose $x^3+x+2$ into irreducible factors in $\mbox{$\mathbb{Z}^{}$}_3[x]$.

Problem 16   Let $p(z)$ be a nonconstant polynomial with real coefficients such that for some real number $a$, $p(a)\neq 0$ but $p'(a)= p''(a)=0$. Prove that the equation $p(z)=0$ has a nonreal root.

Problem 17   Suppose

$$f(z) = \sum_{n=0}^{\infty}a_nz^n$$

has radius of convergence $R>0$. Show that

$$h(z) = \sum_{n=0}^{\infty}\frac{a_nz^n}{n!}$$

is entire and that for $0<r<R$, there is a constant $M$ such that

$$\vert h(z)\vert\leq Me^{\vert z\vert/r}.$$

Problem 18   Show that there is a unique continuous function $f:[0,1]\to\mbox{$\mathbb{R}^{}$}$ such that

$$f(x) = \sin x + \int_0^1\frac{f(y)}{e^{x+y+1}}\,dy.$$

Problem 19   Let $x(t)$ be the solution of the differential equation

$$x''(t)+8x'(t)+25x(t) = 2\cos t$$

with initial conditions $x(0)=0$ and $x'(0)=0$. Show that for suitable constants $\alpha$ and $\delta$,

$$\lim_{t\to\infty}\left(x(t)-\alpha\cos(t-\delta)\right)=0.$$

Problem 20   Evaluate

$$\int_{-\infty}^{\infty}\frac{x\sin x}{x^2+4x+20}\,dx\, .$$