Preliminary Exam - Summer 1977

Problem 1   Prove the following statements about the polynomial ring $\mbox{\bf {F}}[x]$, where $\mbox{\bf {F}}$ is any field.

1. $\mbox{\bf {F}}[x]$ is a vector space over $\mbox{\bf {F}}$.
2. The subset $\mbox{\bf {F}}_n[x]$ of polynomials of degree $\leq n$ is a subspace of dimension $n+1$ in $\mbox{\bf {F}}[x]$.
3. The polynomials $1, x-a, \ldots, (x-a)^n$ form a basis of $\mbox{\bf {F}}_n[x]$ for any $a \in \mbox{\bf {F}}$.

Problem 2   Let f be continuous on $\mathbb{C}\,^{}$ and analytic on $\{z \;\vert\; \Im z \neq 0 \}$. Prove that $f$ must be analytic on $\mathbb{C}\,^{}$.

Problem 3   Prove that $\alpha = \sqrt{3} +\sqrt{5}$ is algebraic over $\mathbb{Q}\,^{}$, by explicitly finding a polynomial with coefficients in $\mathbb{Q}\,^{}$ of which $\alpha$ is a root.

Problem 4   Let $A$ be an $r\times r$ matrix of real numbers. Prove that the infinite sum

$$e^{A} = I + A + \frac{A^2}{2}+\cdots+\frac{A^n}{n!} +\cdots$$

of matrices converges (i.e., for each $i,j$, the sum of $(i,j)^{th}$ entries converges), and hence that $e^{A}$ is a well-defined matrix.

Problem 5   Write all values of $i^i$ in the form $a+bi$.

Problem 6   Show that

$$F(k) = \int_0^{\pi/2}\frac{dx}{\sqrt{1-k\cos^2x}}\;$$

$0\leq k < 1$, is an increasing function of $k$.

Problem 7   Let $A : \mbox{$\mathbb{R}^{6}$} \to \mbox{$\mathbb{R}^{6}$}$ be a linear transformation such that $A^{26} = I$. Show that $\mbox{$\mathbb{R}^{6}$} = V_1\oplus V_2\oplus V_3$, where $V_1$, $V_2$, and $V_3$ are two-dimensional invariant subspaces for $A$.

Problem 8   Prove that the initial value problem

$$\frac{dx}{dt} = 3x+85\cos x, \hspace{.2in} x(0) = 77\,,$$

has a solution $x(t)$ defined for all $t \in \mbox{$\mathbb{R}^{}$}$.

Problem 9   Show that every rotation of $\mathbb{R}^{3}$ has an axis; that is, given a 3$\times$3 real matrix $A$ such that $A^t=A^{-1}$ and $\det A > 0$, prove that there is a nonzero vector v such that Av = v. _matrix,>axis

Problem 10   Suppose that $f(x)$ is defined on $[-1,1]$, and that $f'''(x)$ is continuous. Show that the series

$$\sum_{n=1}^{\infty}\, \left( n \left( f(1/n) - f(-1/n) \right) - 2f'(0)\right)$$

converges.

Problem 11   Let $f(x,t)$ be a $C^1$ function such that $\frac{\partial f}{\partial x} = \frac{\partial f}{\partial t}$. Suppose that $f(x,0)>0$ for all $x$. Prove that $f(x,t)>0$ for all $x$ and $t$.

Problem 12   Let $V$ be the vector space of all polynomials of degree $\leq 10$, and let $D$ be the differentiation operator on $V$ (i.e., $Dp(x)=p'(x)$).

1. Show that $\mathrm{tr}\; D = 0$.

2. Find all eigenvectors of $D$ and $e^D$.

Problem 13   Let $f$ be an analytic function such that $f(z)=1+2z+3z^2+\cdots$ for $\vert z\vert<1$. Define a sequence of real numbers $a_0,a_1,a_2,\ldots$ by

$$f(z) = \sum_{n=0}^{\infty}a_n(z+2)^n.$$

What is the radius of convergence of the series

$$\sum_{n=0}^{\infty}a_nz^n?$$

Problem 14  

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 $\mbox{$\mathbb{Q}\,^{}$}/\mbox{$\mathbb{Z}^{}$}$, where $\mathbb{Z}^{}$ is the group of integers?

Note: See also Problems and .

Problem 15   Let $A\subset\mbox{$\mathbb{R}^{n}$}$ be compact, $x\in A$; let $(x_i)$ be a sequence in $A$ such that every convergent subsequence of $(x_i)$ converges to $x$.

1. Prove that the entire sequence $(x_i)$ converges.
2. Give an example to show that if $A$ is not compact, the result in Part 1 is not necessarily true.

Problem 16   Use the Residue Theorem to evaluate the integral

$$I(a) = \int_0^{2\pi}\frac{d\theta}{a+\cos \theta}$$

where $a$ is real and $a>1$. Explain why the formula obtained for $I(a)$ is also valid for certain complex (nonreal) values of $a$.

Problem 17   In the ring $\mbox{$\mathbb{Z}^{}$}[x]$ of polynomials in one variable over the integers, show that the ideal $\mathfrak{I}$ generated by $5$ and $x^2+2$ is a maximal ideal.

Problem 18   Let $\hat{a}_0+\hat{a}_1z+\cdots+\hat{a}_nz^n$ be a polynomial having $\hat{z}$ as a simple root. Show that there is a continuous function $r : U \to \mbox{$\mathbb{C}\,^{}$}$, where $U$ is a neighborhood of $(\hat{a}_0,\ldots,\hat{a}_n)$ in $\mathbb{C}\,^{n+1}$, such that $r(a_0,\ldots,a_n)$ is always a root of $a_0+a_1z+\cdots+a_nz^n$, and $r(\hat{a}_0,\ldots,\hat{a}_n)=\hat{z}$.

Problem 19   Let p be an odd prime. If the congruence $x^2\equiv -1 \pmod{p}$ has a solution, show that $p\equiv 1 \pmod{4}$.

Problem 20   Determine all solutions to the following infinite system of linear equations in the infinitely many unknowns $x_1,x_2,\ldots$:

$$\begin{array}{ccccccc} x_1 & + &x_3 & + &x_5 & = & 0 \\ x_2 ... ... = & 0 \\ \vdots & & \vdots & & \vdots & & \vdots \end{array}$$

How many free parameters are required?