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\title{Ph.D. Preliminary Examination in Algebra}
\date{January 25, 1999}
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\begin{document}

\begin{enumerate}

\item
Let $G$ be a finite group and $\mbox{Perm}(G)$ the group of permutations
of $G$ viewed as a set.
\begin{enumerate}
\item
Show that the map
\[ \lambda;: \spc G \hsp\longrightarrow\hsp \mbox{Perm}(G) \]
that is defined by \hsp$\lambda(\sigma)(\tau) = \sigma \circ \tau$\hsp
is a group homomorphism.

\item
Show that the map $\rho;_1$ defined by \hsp$\rho;_1(\sigma)(\tau) =
\tau \circ \sigma$\hsp is a homomorphism if and only if $G$ is an
abelian group.

\item
Show that the map $\rho$ defined by \hsp$\rho(\sigma)(\tau) =
\tau \circ \sigma^{-1}$\hsp is a homomorphism for every group $G$.

\end{enumerate}


\item
Let $A$ be an $n \times n$ matrix in a field $K$, \hsp let $c(t)$ be the
\emph{characteristic polynomial} of $A$, \hsp and let $m(t)$ be the
\emph{minimal polynomial} of $A$.  Show that $m(t)$ divides $c(t)$
in the polynomial ring $K[t]$.


\item
Show that the alternating group $A_4$ has no subgroup of index $2$.


\item
Let \hsp;$f(x) = x^5 - 2$\hsp; in $\mbox{\bold{Q}}\lsb;x\rsb;$, and let $K$ be the
splitting field of $f(x)$ over \bold{Q}.
\begin{enumerate}
\item
Find generators for $K$ as a $\mbox{\bold{Q}}$-algebra.

\item
Find the Galois group $G$ of $K$ over \bold{Q}.

\item
For each subgroup $H$ of $G$ describe the subfield of $K$ which corresponds
to $H$ under the \quophrase{fundamental correspondence of Galois theory}.

\end{enumerate}


\item
Show that if a finite ring $R$ admits an injective (ring) homomorphism from
a field,\hsp; then the number of elements of $R$ must be a power of a
prime number.


\item
Let $R$ be a commutative ring, $H$ a commutative $R$-algebra, \hsp and
$I$ an ideal in $H$.  Show that
\[ H/I \hsp\otimes_R\hsp H/I \spc\cong\spc
 \frac{H \hsp\otimes_R\hsp H}{
   I \hsp\otimes_R\hsp H \spc+\spc H \hsp\otimes_R\hsp I} \spc\spc \eos \]


\item
Let the field $L$ be a (finite) Galois extension of the field $K$.
Define \hsp;$\mbox{tr}:\hsp L \hsp\rightarrow\hsp K$\hsp; by
\[ \mbox{tr}(\alpha) \spc = \spc
             \sum_{\sigma\in G} \sigma(\alpha)\sum: \eos \]
Show that this \emph{trace} map is surjective on $K$.


\item
Let $R$ be a ring and $P$ a left $R$-module.  Show that the following
two statements are equivalent:
\begin{enumerate}
\item
$P$ is a direct summand of a finitely-generated free left $R$-module.

\item
There exist \hsp;$x_1, \ldots, x_n \in P$, and \hsp;$f_1, \ldots, f_n
\in \mbox{Hom}_{R}(P, R)$\hsp; such that the relation
\[ x \spc = \spc \sum f_i(x) x_i \sum: \]
holds for all $x \in P$.

\end{enumerate}

\end{enumerate}
\end{document}