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\title{Ph.D. Preliminary Examination in Algebra}
\date{June 9, 2000}
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\begin{document}
\begin{center}\LARGE\bf
Ph.D. Preliminary Examination in Algebra
\end{center}
\begin{center}
\large\bf
June 9, 2000
\end{center}

\medskip
\begin{enumerate}
\item   Let \(X\) be any set, \(F\) any field, and \(F^{X}\) the set of
maps from \(X\) to \(F\)\@. \   \(F^{X}\) is endowed with the structure of
a vector space over \(F\) using ``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 \(\mbox{det}\left(f_{i}(x_{j})\right)\) is non-zero. \ 

\item   Find a complete set of representatives for the isomorphism
classes of finite abelian groups of order~\(1001\)\@. \ 

\item  Let \(K\) be the splitting field over the field \(\mbox{Q}\) of
rational numbers of the polynomial
\[ f(x) = x^{5} - x^{4} + x^{3} - x^{2} + x - 1 \ \ \ . \]
\begin{enumerate}
\item   What are the possible values for the \textbf{minimum} degree among
the irreducible factors of a polynomial of degree \(5\)\@? \ 

\item   Write \(f\) as the product of factors irreducible over \(\mbox{R}\)\@. \ 

\item   Write \(f\) as the product of factors irreducible over \(\mbox{Q}\)\@. \ 

\item   What is the degree of \(K\) over \(\mbox{Q}\)\@? \ 

\item   What is the Galois group of \(K\) over \(\mbox{Q}\)\@? \ 
\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
\[ A = {F[t, x, y, z]}/{(t z - x y) F[t, x, y, z]} \]
where \(t, x, y, z\) are independent transcendentals over \(F\)\@. \ 
\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 \(F\)\@. \ 
\begin{enumerate}
\item   Outline an argument for showing that if \(F\) is a finite
       field, then \(E\) is a cyclic Galois extension of~\(F\)\@. \ 

\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 \(F\)\@. \ 

\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 \(\mbox{\textbf{F}}_{3}\) denote the field of \(3\) elements. \ 
\begin{enumerate}
\item   What is the cardinality of \(2\)-dimensional Cartesian space
       \(\mbox{\textbf{F}}_{3} \times{} \mbox{\textbf{F}}_{3}\) over
       \(\mbox{\textbf{F}}_{3}\)\@? \ 
\item   Let \(N\) denote cardinality of the group
       \(\mbox{GL}_{2}(\mbox{\textbf{F}}_{3})\) of linear automorphisms
       of \(\mbox{\textbf{F}}_{3} \times{} \mbox{\textbf{F}}_{3}\)\@. \  \\
       Compute \(N\)\@. \ 
\item   Observe that the multiplicative group \(\mbox{\textbf{F}}_{3}^{*}\)
       is the unique group of order \(2\) and furthermore that:
       \begin{enumerate}
\item   Multiplication by invertible scalars gives rise
       to a homomorphism \(\phi{}\) from \(\mbox{\textbf{F}}_{3}^{*}\) to
       \(\mbox{GL}_{2}(\mbox{\textbf{F}}_{3})\). \ 
       
\item   The determinant gives rise to a homomorphism \(\psi{}\)
       from \(\mbox{GL}_{2}(\mbox{\textbf{F}}_{3})\) to \(\mbox{\textbf{F}}_{3}^{*}\)
       
\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 \(\phi{}\)\@? \ 
\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}