\documenttype{article}
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\title{Ph.D. Preliminary Examination in Algebra}
\date{August 31, 2001}
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\newcommand{\mx3}[3]{%
\bal{\begin{array}{rrr} #1 \\ #2 \\ #3 \end{array}}}
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\bal{\begin{array}{rrrr} #1 \\ #2 \\ #3 \\ #4 \end{array}}}
\begin{document}

\begin{enumerate}

\item  Let $A$ denote the matrix
\[ \mx3{1 & -2 & 1}{5 & -4 & 3}{3 & -3 & 2} \ ,\]
and let $f$ be the $\Q$-linear endomorphism of the vector space $\Q^{3}$
given by $f(x) = A x$\aos  Find the dimension of the quotient vector
space $\Q^3/\mbox{Image}(f)$\aos

\item  Let $N$ be a normal subgroup of a group $G$ with finite
index $ [G:N] = k $\aos  Show that $g^k \in N$ for each element
$g \in G$\aos

\item  Why must the number of elements in a finite field always be
the power of some prime?

\item  Does the existence of the relationship
\[ \bal{2 + \sqrt{-5}}\bal{2 - \sqrt{-5}} = \bal{3}\bal{3} \]
bear on the question of whether or not the ring
\[ \Z\lsb; t \rsb;/(t^2 + 5)\Z\lsb; t \rsb; \]
is a principal ideal domain?  (In this $\Z$ denotes the ring of
integers, and $\Z\lsb; t \rsb;$ denotes the ring of polynomials in one
variable over $\Z$\aos)  Explain your answer.

\item  Let $M$ be the $4 \times 4$ matrix
\[
\mx4{1 & 0 & 0 & 0}{0 & 0 & 0 & 1}{0 & -1 & 1 & -1}{0 & -1 & 0 & 0}
\ \eos;\]
Find the characteristic and minimal polynomials of $M$ when it is
regarded as a matrix over the field $\C$ of complex numbers.

\item  What is the Galois group of the polynomial $x^4 + 1$ over
the field $\Q$ of rational numbers?

\item  Prove over \emph{any commutative ring} (with $1$) that two
isomorphic free modules of finite rank must have the same rank.

\item  Find the group of all automorphisms of the symmetric group
on $3$ letters.

\end{enumerate}
\end{document}