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\title{Preliminary Examination in Algebra}
\date{August, 2005}
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\begin{document}
\begin{center}
{\LARGE\bfseries{}Preliminary Examination in Algebra}
\\\hsf\\{\large\bfseries{}Department of Mathematics \& Statistics}
\\[4\partopsep]{\large\bfseries{}August, 2005}
\end{center}
\vspace*{-\centerskip}
\par{\textbf{Directions:} \ \ There are 8 questions, all of the same weight. \ 
Please take the time to ensure accuracy and completeness, especially
for the questions you find easiest. \   (Completeness does not mean
excessive verbosity. \   You should not attempt to prove standard
propositions that you cite except where the proof of a standard
proposition is explicitly sought.)
}
\par{The ring of integers will be denoted by \(\mathbb{Z}\) and its field of fractions
by \(\mathbb{Q}\)\@. \ 

\begin{enumerate}
%nul
\item  Prove that a non-abelian group of order \(2p\), \(p\) an odd prime, must
have a trivial center. \ 

%nul

\item  When \(F\) is a field, let \(\mbox{GL}_{n}(F)\) denote the group of all
invertible \(n \times{} n\) matrices in \(F\) under the operation of matrix
multiplication, and let \(\mbox{SL}_{n}(F)\) denote its subgroup defined by
restricting to matrices of determinant \(1\). \   Find a subgroup \(H\) of
\(\mbox{GL}_{n}(F)\) such that \(\mbox{GL}_{n}(F)\) is
isomorphic to the semi-direct product of \(H\) with \(\mbox{SL}_{n}(F)\). \ 

%nul

\item  Prove that the number of elements in any finite field must be a
prime power. \ 

%nul

\item  Let \(\mathbb{Z}/m\mathbb{Z}\) denote the ring of integers modulo \(m\). \   Let \(r,s\) be
positive integers. \ 
  \begin{enumerate}

\item  What element of \(\mathbb{Z}\) generates the ideal \(r\mathbb{Z} + s\mathbb{Z}\)? \ 
  

\item  What is the kernel of the canonical ring homomorphism
         \(\mathbb{Z}/rs\mathbb{Z} \longrightarrow{} \mathbb{Z}/r\mathbb{Z} \times{} \mathbb{Z}/s\mathbb{Z}\)? \ 
  

\item  Find an integer \(t\) such that
         \(\mathbb{Z}/r\mathbb{Z} \otimes  \mathbb{Z}/s\mathbb{Z} \cong{} \mathbb{Z}/t\mathbb{Z}\). \ 
  
\end{enumerate}

%nul

\item  Let \(M\) be a \(3 \times{} 3\) matrix over the rational field \(\mathbb{Q}\)
whose characteristic polynomial is
\[ t^{3} + 2t^{2} - 4t - 8 \ \ \ . \]
Find:
  \begin{enumerate}

\item  all possible sequences of (polynomial) invariant factors
         for \(M\). \ 
  

\item  representatives of the different possible similarity
         classes of such matrices \(M\). \ 
  
\end{enumerate}

%nul

\item  For any integer \(n \geq{} 3\) let \(D_{n}\) denote the \(n\)th dihedral group,
i.e., the group of order \(2n\) that is the semi-direct product of the
cylic group \(\mathbb{Z}/n\mathbb{Z}\) with \(\mathbb{Z}/2\mathbb{Z}\) for the unique non-trivial
action (by automorphisms) of the latter on the former, or, equivalently,
the group of symmetries of a regular \(n\)-gon. \ 
  \begin{enumerate}

\item  Describe \(D_{n}\) by generators and relations. \ 
  

\item  Show that every automorphism of the dihedral group \(D_{3}\) is
         inner, i.e., is the conjugation by some element of \(D_{3}\). \ 
  

\item  Show that for any \(n\) odd, \(n \geq{} 5\), the dihedral group \(D_{n}\)
         has an automorphism that is not inner. \ 
  
\end{enumerate}

%nul

\item  For any integer \(n > 1\), explain how to find a field \(K\) and a
polynomial \(f(x)\in{} K [x]\) of degree \(n\) so that \(L \, = \, K [x]/(f(x))\) is
a Galois field extension of \(K\) with cyclic Galois group of order \(n\). \ 

%nul

\item  Let \(\mathbb{F}_{p}\) denote the field \(\mathbb{Z}/p\mathbb{Z}\) of \(p\) elements. \   In
\(\mbox{M}_{n}(\mathbb{F}_{p})\), the ring of \(n \times{} n\) matrices with entries
in \(\mathbb{F}_{p}\), let \(N\) be the \(n \times{} n\) Jordan block matrix with
\(1\)'s on the first superdiagonal (and \(0\)'s everywhere else including
the main diagonal). \   Let \(\mathbb{F}_{p} [N]\) be the \(\mathbb{F}_{p}\)-algebra of
polynomials with coefficients in \(\mathbb{F}_{p}\) evaluated at \(N\),  let \(U\)
be the group of units in \(\mathbb{F}_{p} [N] \). \ 
  \begin{enumerate}

\item  Describe the elements of \(U\). \ 
  

\item  Find the exponent of \(U\) when \(p > n\). \ 
  

\item  Describe the isomorphism type of \(U\) as a finite abelian group
         when \(p > n\). \ 
  
\end{enumerate}

\end{enumerate}

}
\end{document}

