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\title{Ph.D. Preliminary Examination in Algebra}
\date{August 31, 2001}
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\begin{document}
\begin{center}\LARGE\bf
Ph.D. Preliminary Examination in Algebra
\end{center}
\begin{center}
\large\bf
August 31, 2001
\end{center}

\medskip
\begin{enumerate}
\item  Let \(A\) denote the matrix
\[ \left(\begin{array}{rrr}
1 & -2& 1\\{}
5& -4& 3\\{}
3& -3& 2\end{array}\right) \ ,\]
and let \(f\) be the \(\mbox{\textbf{Q}}\)-linear endomorphism of the vector space \(\mbox{\textbf{Q}}^{3}\)
given by \(f(x) \, = \, A x\)\@. \   Find the dimension of the quotient vector
space \(\mbox{\textbf{Q}}^{3}/\mbox{Image}(f)\)\@. \ 

\item  Let \(N\) be a normal subgroup of a group \(G\) with finite
index \( [G:N] \, = \, k \)\@. \   Show that \(g^{k} \in{} N\) for each element
\(g \in{} G\)\@. \ 

\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
\[ \left(2 + \sqrt{-5}\right)\left(2 - \sqrt{-5}\right) \ = \  \left(3\right)\left(3\right) \]
bear on the question of whether or not the ring
\[ \mbox{\textbf{Z}}[ t ]/(t^{2} + 5)\mbox{\textbf{Z}}[ t ] \]
is a principal ideal domain? \   (In this \(\mbox{\textbf{Z}}\) denotes the ring of
integers, and \(\mbox{\textbf{Z}}[ t ]\) denotes the ring of polynomials in one
variable over \(\mbox{\textbf{Z}}\)\@. \ )  Explain your answer. \ 

\item  Let \(M\) be the \(4 \times{} 4\) matrix
\[\left(\begin{array}{rrrr}
1 & 0& 0& 0\\{}
0& 0& 0& 1\\{}
0& -1& 1& -1\\{}
0& -1& 0& 0\end{array}\right)
\ \ \ .\]
Find the characteristic and minimal polynomials of \(M\) when it is
regarded as a matrix over the field \(\mbox{\textbf{C}}\) of complex numbers. \ 

\item  What is the Galois group of the polynomial \(x^{4} + 1\) over
the field \(\mbox{\textbf{Q}}\) 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}