\documentclass{article}[12pt]
\usepackage{amssymb}
\pagestyle{empty}
\begin{document}
\pagestyle{empty}
\large
\begin{center} Mathematics Department \hspace{.5in} March 29, 1997 \end{center}
\begin{center} Ph.D. Qualifying Exam \end{center}
\begin{center} ALGEBRA \end{center}
\begin{center} March 29, 1997 \end{center}
\bigskip
\begin{quote}
\noindent INSTRUCTIONS: Do any four problems. And no more than
four.\\
Please make sure that you give complete solutions to each problem
that you do.\\
{\bf Indicate which problems you wish to have graded.}
You have three hours.
\vspace{.25in}
POLICY ON MISPRINTS\\
The Ph.D. Qualifying Examination Committee tries to proofread the
exams as carefully as possible. Nevertheless, the exam may
contain misprints. If you are convinced a problem has been stated
incorrectly, mention this to the proctor and indicate your
interpretation in your solution. In such cases do not interpret
the problem in such a way that it becomes trivial.
\end{quote}
\newpage
\begin{enumerate}
\item Let $G$ be a finite group, $p$ a prime, and $P$ a
$p$-subgroup of $G$. Let $n$ be the number of Sylow $p$-subgroups
of $G$ that contain $P$.\
(a) [4 points] Prove $n \equiv 1$ mod $p$.
(b) [4 points] Prove $n \ge |Syl_p(N_G(P)|$.
(c) [2 points] Prove that $G = S_4$ (the symmetric group on 4
letters) is an example of strict inequality in part (b).\\
\item Let $G$ be a finite simple group of order 168.\
(a) [2 points] Show that $G$ has precisely 8 Sylow 7-subgroups.
(b) [3 points] Show that $G$ is isomorphic to a subgroup of
$\tilde{G}$ of $A_8$ and that no element of order 2 in
$\tilde{G}$ has a fixed point.
(c) [2 points] Show that $G$ has no element of order 6.
(d) [3 points] Find the number of Sylow 3-subgroups of $G$.\\
\item Find an orthogonal matrix $Q$ and a diagonal matrix
$\Lambda$ so that $Q^{-1}AQ = \Lambda$ where $$\left(
\begin{array}{ccc}
1 & -2 & 0\\
-2 & 0 & 2\\
0 & 2 & -1 \end{array} \right) $$\\
\item Let $R$ be a ring and $M$ be an $R$-module of finite
composition length. If $f$ is an endomorphism of the $R$-module
$M$, show that there is an integer $k$ such that $M$ = Im $f^k
\oplus$ Ker $f^k$.\\
\item If $K = {\Bbb Q}(\sqrt a)$, where $a$ is a negative integer
and ${\Bbb Q}$ is the rationals, show that $K$ can't be embedded
in a cyclic extension whose degree over ${\bf Q}$ is divisible by
4.\\
\item Let $p$ be a prime and let $GF(p^m)$ denote a finite field
of order $p^m$.
(a) [5 points] Show that $GF(p^m)$ is isomorphic to a subfield of
$GF(p^n)$ if and only if $m$ divides $n$.
(b) [5 points] Let $E$ be the algebraic closure of $GF(p)$. Show
that there is an intermediate field $L$ between $GF(p)$ and $E$
with $|L:GF(p)| = \infty$ and $|E:L| = \infty$.\\
\item Let $R$ be a commutative ring with 1 and $R[x]$ the ring of
polynomials in one (commuting) indeterminate with coefficients in
$R$.
(a) [8 points] For each of the following statements indicate
whether it is true or false. If it is false, give a
counterexample. If it is true you do NOT have to provide a proof.
(i) If $R$ is a PID then so is $R[x]$.
(ii) If $R$ is a UFD then so is $R[x]$.
(iii) If $R$ is artinian then so is $R[x]$.
(iv) If $R$ is noetherian then so is $R[x]$.\\
(b) [2 points] What are the units in $R[x]$? Justify your
answer.\\
\item Let $R$ be a ring with 1. If $M$ is an $R$-module, the {\bf
uniform dimension} of $M$ (ud $M$) is the largest integer $n$
such that there is a direct sum $M_1 \oplus \ldots \oplus M_n
\subseteq M$ with all the $M_i$ non-zero. If no such integer
exists then we say ud $M = \infty$. If $M \subseteq N$ are
$R$-modules, $M$ is said to be {\bf essential} in $N$ if every
non-zero submodule of $N$ has non-zero intersection with $M$.
Suppose the ud $M < \infty$ and $M \subseteq N$. Prove that $M$
is essential in $N$ if and only if ud $M$ = ud $n$.
\end{enumerate}
\end{document}