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

\medskip
\begin{description}
\item[{ 1. \ }]
{Let \(G\) be a finite group and \(\mbox{Perm}(G)\) the group of permutations
of \(G\) viewed as a set. \ 
\begin{description}
\item[{ (a) \ }]
{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[{ (b) \ }]
{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[{ (c) \ }]
{Show that the map \(\rho{}\) defined by \,\(\rho{}(\sigma{})(\tau{}) =
\tau{} \circ  \sigma{}^{-1}\)\, is a homomorphism for every group \(G\). \ }
\end{description}}
\item[{ }]
{ }
\item[{ 2. \ }]
{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[{ }]
{ }
\item[{ 3. \ }]
{Show that the alternating group \(A_{4}\) has no subgroup of index \(2\)\@. \ }
\item[{ }]
{ }
\item[{ 4. \ }]
{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{description}
\item[{ (a)\  }]
{Find generators for \(K\) as a \(\mbox{\textbf{Q}}\)-algebra. \ }
\item[{ (b)\  }]
{Find the Galois group \(G\) of \(K\) over \textbf{Q}. \ }
\item[{ (c)\  }]
{For each subgroup \(H\) of \(G\) describe the subfield of \(K\) which corresponds
to \(H\) under the ``fundamental correspondence of Galois theory''. \ }
\end{description}}
\item[{ }]
{ }
\item[{ 5. \ }]
{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[{ }]
{ }
\item[{ 6. \ }]
{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[{ }]
{ }
\item[{ 7. \ }]
{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[{ }]
{ }
\item[{ 8. \ }]
{Let \(R\) be a ring and \(P\) a left \(R\)-module. \   Show that the following
two statements are equivalent:
\begin{description}
\item[{ (a) \ }]
{\(P\) is a direct summand of a finitely-generated free left \(R\)-module. \ }
\item[{ (b) \ }]
{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{description}}
\end{description}\end{document}