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\title{Ph.D. Preliminary Examination in Algebra}
\date{January 25, 1999}
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\begin{document}
\begin{center}\LARGE\bfseries{}
Ph.D. Preliminary Examination in Algebra
\end{center}
\begin{center}
\large\bfseries{}
January 25, 1999
\end{center}

\medskip
\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 %
\[ \lambda{}: \  G \,\longrightarrow{}\, \mbox{Perm}(G) \] %
that is defined by \,\(\lambda{}(\sigma{})(\tau{}) \, = \, \sigma{} \circ  \tau{}\)\, %
is a group homomorphism. \  %


\item  Show that the map \(\rho{}_{1}\) defined by \,\(\rho{}_{1}(\sigma{})(\tau{}) \, = \, %
\tau{} \circ  \sigma{}\)\, is a homomorphism if and only if \(G\) is an %
abelian group. \  %


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

\end{enumerate} %
 %


\item  Let \(A\) be an \(n \times{} n\) matrix in a field \(K\), \, let \(c(t)\) be the %
\emph{characteristic polynomial} of \(A\), \, 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 \,\(f(x) \, = \, x^{5} - 2\)\, in \(\mbox{\textbf{Q}}[x]\), and let \(K\) be the %
splitting field of \(f(x)\) over \textbf{Q}. \  %
\begin{enumerate}

\item  Find generators for \(K\) as a \(\mbox{\textbf{Q}}\)-algebra. \  %


\item  Find the Galois group \(G\) of \(K\) over \textbf{Q}. \  %


\item  For each subgroup \(H\) of \(G\) describe the subfield of \(K\) which corresponds %
to \(H\) under the ``fundamental correspondence of Galois theory''. \  %

\end{enumerate} %
 %


\item  Show that if a finite ring \(R\) admits an injective (ring) homomorphism from %
a field,\, 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, \, and %
\(I\) an ideal in \(H\). \   Show that %
\[ H/I \,\otimes _{R}\, H/I \ \cong{}\  %
 \frac{H \,\otimes _{R}\, H}{   I \,\otimes _{R}\, H \ +\  H \,\otimes _{R}\, I} \ \  \ \ . \] %
 %


\item  Let the field \(L\) be a (finite) Galois extension of the field \(K\). \  %
Define \,\(\mbox{tr}:\, L \,\rightarrow{}\, K\)\, by %
\[ \mbox{tr}(\alpha{}) \  \ = \  \  %
             \sum_{\sigma{}\in{} G} \sigma{}(\alpha{}) \ \ . \] %
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 \,\(x_{1}, \ldots{}, x_{n} \in{} P\), and \,\(f_{1}, \ldots{}, f_{n} %
\in{} \mbox{Hom}_{R}(P, R)\)\, such that the relation %
\[ x \  \ = \  \  \sum f_{i}(x) x_{i}  \] %
holds for all \(x \in{} P\). \  %

\end{enumerate} %

\end{enumerate}\end{document}

