Preliminary Exam - Spring 1996

Problem 1   Compute

$$L = \lim_{n\to \infty} \left( n^n \over n! \right)^{1/n}.$$

Problem 2   Let $K \subset \mbox{$\mathbb{R}^{n}$}$ be compact and $\lbrace B_j \rbrace_{j=1}^\infty$ be a sequence of open balls which covers $K$. Prove that there is $\varepsilon > 0$ such that every $\varepsilon$-ball centered at a point of $K$ is contained in one of the balls $B_j$.

Problem 3   Compute

$$I = \int_0^{2\pi} {d\theta \over 2+\cos \theta} .$$

Problem 4   Let $r < 1 < R$. Show that for all sufficiently small $\varepsilon >0$, the polynomial

$$p(z)= \varepsilon z^7 + z^2 + 1$$

has exactly five roots (counted with their multiplicities) inside the annulus

$$r\varepsilon^{-1/5} < \vert z\vert < R \varepsilon^{-1/5}.$$

Problem 5   Prove or disprove: For any 2$\times$2 matrix $A$ over $\mbox{$\mathbb{C}\,^{}$}$, there is a 2$\times$2 matrix $B$ such that $A=B^2$.

Problem 6   If a finite homogeneous system of linear equations with rational coefficients has a nonzero complex solution, need it have a nonzero rational solution? Prove or give a counterexample.

Problem 7   Prove that $f(x)=x^4+x^3+x^2+6x+1$ is irreducible over $\mbox{$\mathbb{Q}\,^{}$}$.

Problem 8   Determine the rightmost decimal digit of

$$A = 17^{17^{17}}.$$

Problem 9   Exhibit infinitely many pairwise nonisomorphic quadratic extensions of $\mbox{$\mathbb{Q}\,^{}$}$ and show they are pairwise nonisomorphic.

Problem 10   Show that a positive constant $t$ can satisfy

$$e^x > x^t \quad for \; all \quad x > 0$$

if and only if $t<e$.

Problem 11   Suppose $\varphi$ is a $C^1$ function on $\mbox{$\mathbb{R}^{}$}$ such that

$$\varphi(x) \to a \quad and \quad \varphi^\prime(x) \to b \quad as \quad x \to \infty.$$

Prove or give a counterexample: $b$ must be zero.

Problem 12   Let $M_{2 \times 2}$ be the space of 2$\times$2 matrices over $\mbox{$\mathbb{R}^{}$}$, identified in the usual way with $\mbox{$\mathbb{R}^{4}$}$. Let the function $F$ from $M_{2 \times 2}$ into $M_{2 \times 2}$ be defined by

$$F(X) = X + X^2.$$

Prove that the range of $F$ contains a neighborhood of the origin.

Problem 13   Let $f=u+iv$ be analytic in a connected open set $D$, where $u$ and $v$ are real valued. Suppose there are real constants $a$, $b$ and $c$ such that $a^2+b^2 \not= 0$ and

$$au+bv=c$$

in $D$. Show that $f$ is constant in $D$.

Problem 14   Suppose $f\colon [0,1]\to \mbox{$\mathbb{C}\,^{}$}$ is continuous. Show that

$$g(z) = \int_0^1 f(t) e^{tz^2} dt$$

defines a function $g$ that is analytic everywhere in the complex plane.

Problem 15   Suppose that $A$ and $B$ are real matrices such that $A^t=A$,

$$v^t A v \ge 0$$

for all $v \in \mbox{$\mathbb{R}^{n}$}$ and

$$AB + BA = 0.$$

Show that $AB=BA=0$ and give an example where neither $A$ nor $B$ is zero.

Problem 16   Let $A$ be the $n \times n$ matrix which has zeros on the main diagonal and ones everywhere else. Find the eigenvalues and eigenspaces of $A$ and compute $\det(A)$.

Problem 17   Let $G$ be the group of 2$\times$2 matrices with determinant $1$ over the four-element field $\mbox{\bf {F}}$. Let $S$ be the set of lines through the origin in $\mbox{\bf {F}}^2$. Show that $G$ acts faithfully on $S$. (The action is faithful if the only element of $G$ which fixes every element of $S$ is the identity.) _group action, faithful

Problem 18   Let $G$ and $H$ be finite groups of relatively prime orders. Show that the automorphism group $\mathrm{Aut}(G\times H)$ is isomorphic to the direct product of $\mathrm{Aut}(G)$ and $\mathrm{Aut}(H)$.