\documenttype{article}
% This is a GELLMU file used to generate the HTML and LaTeX versions.
% \base{href="file://localhost/math/local/dept/prelims/hammond/"}
\surtitle{}
\title{Ph.D. Preliminary Examination in Algebra}
\date{June 9, 2000}
\nogratlinks
\nobanner
\latexcommand{\bsl;setlength\{\bsl;topmargin\}\{0bp\}}
\latexcommand{\bsl;setlength\{\bsl;headsep\}\{-18bp\}}
\latexcommand{\bsl;setlength\{\bsl;textheight\}\{684bp\}}
\begin{document}

\begin{enumerate}

\item  Let $X$ be any set, $F$ any field, and $F^X$ the set of
maps from $X$ to $F$\aos;  $F^X$ is endowed with the structure of
a vector space over $F$ using \quophrase{pointwise} addition
and multiplication by scalars.  Prove that a finite sequence
$f_1, f_2, \ldots, f_n$ of elements of $F^X$ is linearly independent
if and only if there is a finite sequence of elements
$x_1, x_2, \ldots, x_n$ in $X$ for which the $n \times n$
determinant $\mbox{det}\bal{f_i(x_j)}$ is non-zero.

\item  Find a complete set of representatives for the isomorphism
classes of finite abelian groups of order~$1001$\aos

\item Let $K$ be the splitting field over the field $\mbox{Q}$ of
rational numbers of the polynomial
\[ f(x) = x^5 - x^4 + x^3 - x^2 + x - 1 \ \eos; \]
\begin{enumerate}

\item  What are the possible values for the \bold{minimum} degree among
the irreducible factors of a polynomial of degree $5$\aoq

\item  Write $f$ as the product of factors irreducible over $\mbox{R}$\aos

\item  Write $f$ as the product of factors irreducible over $\mbox{Q}$\aos

\item  What is the degree of $K$ over $\mbox{Q}$\aoq

\item  What is the Galois group of $K$ over $\mbox{Q}$\aoq
\end{enumerate}

\item  Show that if two square matrices of the same finite size over a
field are similar in a larger field then they must be similar in the
original field.

\item  Let $F$ be a field, and let $A$ be the quotient ring
\[ A = {F\lsb;t, x, y, z\rsb;}/{(t z - x y) F\lsb;t, x, y, z\rsb;} \]
where $t, x, y, z$ are independent transcendentals over $F$\aos
\begin{enumerate}
\item  Show that $A$ has no zero divisors.
\item  Explain briefly why $A$ is Noetherian.
\item  Is $A$ a unique factorization domain?  (Either prove that it
       is or exhibit an example of something that does not factor
       uniquely according to the usual criteria for such uniqueness.)
\end{enumerate}

\item  Let $E$ be a finite extension of a field $F$\aos
\begin{enumerate}

\item  Outline an argument for showing that if $F$ is a finite
       field, then $E$ is a cyclic Galois extension of~$F$\aos

\item  Provide an example where $F$ is a field of characteristic
       $5$ and $E$ is an extension of $F$ of degree $5$ that is
       not a Galois extension of $F$\aos

\item  For any given field $K$ explain how to obtain an extension $F$
       of $K$ and a finite extension $E$ of $F$ for which
       $E$ is a Galois extension of $F$ with Galois group isomorphic
       to the symmetric group $S_n$ (consisting of the permutations
       of $n$ objects).
\end{enumerate}

\item  Let $\mbox{\bold{F}}_3$ denote the field of $3$ elements.
\begin{enumerate}
\item  What is the cardinality of $2$-dimensional Cartesian space
       $\mbox{\bold{F}}_3 \times \mbox{\bold{F}}_3$ over
       $\mbox{\bold{F}}_3$\aoq
\item  Let $N$ denote cardinality of the group
       $\mbox{GL}_2(\mbox{\bold{F}}_3)$ of linear automorphisms
       of $\mbox{\bold{F}}_3 \times \mbox{\bold{F}}_3$\aos\brk;
       Compute $N$\aos
\item  Observe that the multiplicative group $\mbox{\bold{F}}_3^{*}$
       is the unique group of order $2$ and furthermore that:
       \begin{enumerate}
       \item  Multiplication by invertible scalars gives rise
       to a homomorphism $\phi$ from $\mbox{\bold{F}}_3^{*}$ to
       $\mbox{GL}_2(\mbox{\bold{F}}_3)$.
       \item  The determinant gives rise to a homomorphism $\psi$
       from $\mbox{GL}_2(\mbox{\bold{F}}_3)$ to $\mbox{\bold{F}}_3^{*}$
       \end{enumerate}
       Explain why the kernel of $\psi$ and the cokernel of $\phi$ both
       have the same cardinality.
\item  Is the kernel of $\psi$ isomorphic to the cokernel of $\phi$\aoq 
\end{enumerate}

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

\end{enumerate}
\end{document}