# Preliminary Exam - Spring 1996

Problem 1 Compute

$$
L = \operatorname* { l i m } _ { n \to \infty } \left( \frac { n ^ { n } } { n ! } \right) ^ { 1 / n } .
$$

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

Problem 3 Compute

$$
I = \int _ { 0 } ^ { 2 \pi } \frac { d \theta } { 2 + \cos \theta } .
$$

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

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

Problem 5 Prove or disprove: For any ${ \it 2 } \times { \it 2 }$ matrix A over C , there is a 2×2 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 $f ( x ) = x ^ { 4 } + x ^ { 3 } + x ^ { 2 } + 6 x + 1$ is irreducible over $\mathbb { Q }$

Problem 8 Determine the rightmost decimal digit of

$$
A = 1 7 ^ { 1 7 ^ { 1 7 } } .
$$

Problem 9 Exhibit infinitely many pairwise nonisomorphicquadratic extensions $o f \mathbb { Q }$ and show they are pairwise nonisomorphic.

Problem 10 Show that a positive constant t can satisfy

$$
e ^ { x } > x ^ { t } ~ f o r ~ a l l ~ x > 0
$$

if and only $i f t < e .$

Problem 11 Suppose ϕ is a $C ^ { 1 }$ function on R such that

$$
\varphi ( x )  a \quad a n d \quad \varphi ^ { \prime } ( x )  b \quad a s \quad x  \infty .
$$

Prove or give a counterexample: b must be zero.

Problem 12 Let $M _ { 2 \times 2 }$ be the space of 2×2 matrices over $\mathbb { R }$ , identified in the usual way with $\mathbb { R } ^ { 4 }$ . 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 $f = u + i v$ 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 $a ^ { 2 } + b ^ { 2 } \neq 0$ and

$$
a u + b v = c
$$

in D. Show that f is constant in D.

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

$$
g ( z ) = \int _ { 0 } ^ { 1 } f ( t ) e ^ { t z ^ { 2 } } d t
$$

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

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

$$
v ^ { t } A v \geqslant 0
$$

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

$$
A B + B A = 0 .
$$

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

Problem 16 Let A be the n×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 2×2 matrices with determinant 1 over the four-element field F. Let S be the set of lines through the origin in $\mathbf { F } ^ { 2 }$ 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.)

Problem 18 Let G and H be finite groups of relatively prime orders. Show that the automorphism group Aut(G × H) is isomorphic to the direct product of Aut(G) and Aut(H).