\documenttype{article}
\surtitle{Dept of Mathematics \& Statistics: Algebra Prelims}
\title{Preliminary Examination in Algebra}
\subtitle{Department of Mathematics \& Statistics}
\date{August, 2005}
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\begin{document}

\bold{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.)

The ring of integers will be denoted by $\Z$ and its field of fractions
by $\Q$\aos;

\begin{enumerate}

%G
\item
Prove that a non-abelian group of order $2p$, $p$ an odd prime, must
have a trivial center.

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

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

%R
\item
Let $\Zmod{m}$ denote the ring of integers modulo $m$.  Let $r,s$ be
positive integers.
  \begin{enumerate}
  \item  What element of $\Z$ generates the ideal $r\Z + s\Z$?
  \item  What is the kernel of the canonical ring homomorphism
         $\Zmod{rs} \longrightarrow \Zmod{r} \times \Zmod{s}$?
  \item  Find an integer $t$ such that
         $\Zmod{r} \otimes \Zmod{s} \cong \Zmod{t}$.
  \end{enumerate}

%L
\item
Let $M$ be a $3 \times 3$ matrix over the rational field $\Q$
whose characteristic polynomial is
\[ t^3 + 2t^2 - 4t - 8 \ \eos; \]
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}

%G
\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 $\Zmod{n}$ with $\Zmod{2}$ 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}

%F
\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 $n$.

%R
\item  Let $\FF{p}$ denote the field $\Zmod{p}$ of $p$ elements.  In
$\mbox{M}_n(\FF{p})$, the ring of $n \times n$ matrices with entries
in $\FF{p}$, 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 $\FF{p} [N]$ be the $\FF{p}$-algebra of
polynomials with coefficients in $\FF{p}$ evaluated at $N$,  let $U$
be the group of units in $\FF{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}