Preliminary Exam - Spring 1982

Problem 1   Prove the Fundamental Theorem of Algebra: Every nonconstant polynomial with complex coefficients has a complex root. _Fundamental Theorem>of Algebra

Problem 2   Let $S \subset \mbox{$\mathbb{R}^{n}$}$ be a subset which is uncountable. Prove that there is a sequence of distinct points in $S$ converging to a point of $S$.

Problem 3   Let $A$ and $B$ be n$\times$n complex matrices. Prove that

$$\vert\mathrm{tr}(AB^*)\vert^2 \leq \mathrm{tr}(AA^*) \mathrm{tr}(BB^*).$$

Problem 4   Let $f:\mbox{$\mathbb{R}^{2}$} \to \mbox{$\mathbb{R}^{}$}$ have directional derivatives in all directions at the origin. Is $f$ differentiable at the origin? Prove or give a counterexample.

Problem 5   Let $\{g_n\}$ be a sequence of twice differentiable functions on $[0,1]$ such that $g_n(0)=g_n'(0)=0$ for all n. Suppose also that
$\vert g_n''(x)\vert \leq 1$ for all n and all $x \in [0,1]$. Prove that there is a subsequence of $\{g_n\}$ which converges uniformly on $[0,1]$.

Problem 6   Suppose that $f(x)$ is a polynomial with real coefficients and $a$ is a real number with $f(a)\neq 0$. Show that there exists a real polynomial $g(x)$ such that if we define $p$ by $p(x)=f(x)g(x)$, we have $p(a)=1$, $p'(a)=0$, and $p''(a)=0$.

Problem 7   Suppose that the group $G$ is generated by elements $x$ and $y$ that satisfy $x^{5}y^{3}=x^{8}y^{5}=1$. Is $G$ the trivial group?

Problem 8   Find

$$\int_{-\infty}^{\infty}\frac{\cos x}{x^4+1}\,dx$$

by contour integration.

Problem 9   Find the Jordan Canonical Form for the matrix (over $\mathbb{R}^{}$)

$$\left( \begin{array}{ccc} 4 & 1 & 0 \\ -4 & 0 & 0 \\ 19 & 17 & 5 \end{array} \right).$$

Problem 10   Prove that any group of order $77$ is cyclic.

Problem 11   Decide, without too much computation, whether a finite limit

$$\lim_{z\to 0}\left( (\tan z)^{-2} - z^{-2} \right)$$

exists, where z is a complex variable, and if yes, compute the limit.

Problem 12   Prove or give a counterexample: Every connected, locally pathwise connected set in $\mathbb{R}^{n}$ is pathwise connected.

Problem 13   Let $T:V \to W$ be a linear transformation between finite-dimensional vector spaces. Prove that

$$\dim (\ker T) + \dim (\mathrm{range}\,T) = \dim V\, .$$

Problem 14   Let $f:I \to \mbox{$\mathbb{R}^{}$}$ (where $I$ is an interval of $\mathbb{R}^{}$) be such that $f(x)>0,\;x \in I$. Suppose that $e^{cx}f(x)$ is convex in $I$ for every real number c. Show that $\log f(x)$ is convex in $I$.

Note: A function $g:I \to \mbox{$\mathbb{R}^{}$}$ is convex if

$$g \left( tx + (1-t)y \right) \leq tg(x) + (1-t)g(y)$$

for all $x$ and $y$ in $I$ and $0\leq t \leq 1$. _function,>convex

Problem 15   How many nonsingular 2$\times$2 matrices are there over the field of p elements?

Problem 16   Prove that if $G$ is a group containing no subgroup of index $2$, then any subgroup of index $3$ is normal.

Problem 17   Let $\{f_n\}$ be a sequence of continuous functions from $[0,1]$ to $\mbox{$\mathbb{R}^{}$}$. Suppose that $f_n(x) \to 0$ as $n \to \infty$ for each $x \in [0,1]$ and also that, for some constant $K$, we have

$$\left\vert \int_0^1f_n(x)\,dx \right\vert \leq K < \infty$$

for all n. Does

$$\lim_{n\to\infty}\int_0^1f_n(x)\,dx = 0 \, ?$$

Problem 18   For $\Re z \geq 0$, define

$$F(z) = \int_0^{\infty}\frac{e^{-zt}}{1+t^4}\,dt.$$

Show that F(z) is continuous for $\Re z\geq 0$ and analytic for $\Re z>0$.

Problem 19   Show that the initial value problem

$$y'(x) = 2 + 3\sin \left( y(x) \right), \quad y(0) = 4$$

has a solution defined for $-\infty < x < \infty$.

Problem 20   Prove that the polynomial $x^4 + x + 1$ is irreducible over $\mathbb{Q}\,^{}$.