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\title{Ph.D. Preliminary Examination in Algebra}
\date{June 4, 1999}
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\begin{document}

\begin{enumerate}

\item  Let $A$ be an $n \times n$ matrix with entries in the field
$\mbox{\bold{C}}$ of complex numbers that satisfies the relation
\(A^2 = A \).   Show that $A$ is similar to a diagonal
matrix which has only $0$'s and $1$'s along the diagonal.

\item  Furnish examples of the following:
\begin{enumerate}
\item  A finite group that is solvable but not abelian.
\item  A finite group whose center is a proper subgroup of order $2$\aos
\item  A nested sequence of finite groups \(G, \hsp H, \hsp K\) with
       $H$ a normal subgroup of $G$ and $K$ a normal subgroup of $H$
       such that $K$ is not a normal subgroup of $G$\aos
\end{enumerate}

\item  Let $p$ be the polynomial
\(p(t)  =  t^5  +  t^2 +  1\)
regarded as an element of the ring \(A = \mbox{\bold{F}}_2 [t]\)
of polynomials with coefficients in the field \(\mbox{\bold{F}}_2\) of
two elements.  Show that $p$ is irreducible, and then find a
polynomial of degree at most $4$ with the property that its residue
class modulo the ideal $p A$ generates the entire multiplicative group
of units in the quotient ring \(A / p A \).

\item  Let $G$ be a finite group of order $N$\aoc\hsp and let $n$ be a
positive integer that divides $N$\aos  Do \bold{one} of the following:
\begin{enumerate}
\item  Prove that if $G$ is abelian, then $G$ contains a subgroup of
       order $n$\aos
\item  Find an example of \(G, \hsp N, \hsp n\) as above where $G$
       has no subgroup of order $n$\aos
\end{enumerate}

\item  Show that every group of order $30$ contains a normal cyclic
subgroup of order $15$\aos

\item  Let $F$ be the field \(\mbox{\bold{Q}}(i)\) where
\( i = \sqrt{-1} \in \mbox{\bold{C}}\),\spc and let $E$ be the splitting
field over $F$ of the polynomial
\( f(t) = t^4 - 5 \).   Find:
\begin{enumerate}
\item  the extension degree \hsp$[E:F]$\aos
\item  the group \(\mbox{Aut}_F(E)\) of all automorphisms of
       $E$ that fix $F$\aos
\end{enumerate}

\item  Let \(\mbox{\bold{F}}_2\) be the field of $2$ elements, and let
$R$ be the commutative ring
\[ R \spc = \spc \mbox{\bold{F}}_2 [t]/t^3 \mbox{\bold{F}}_2 [t] \hsp . \]
\begin{enumerate}
\item  How many elements does $R$ contain?
\item  What is the characteristic of $R$\aoq
\item  Find all ring homomorphisms
           \( R \hsp \rightarrow \hsp R \). 
\end{enumerate}

\item  Let \(a, b, c, d\) be elements of a field $F$\aoc\hsp let
\(A, B, C, D\) be $n \times n$ matrices over $F$\aoc\hsp and let
\[
m \spc=\spc\bal{\begin{mtable}{rr}\tr\td a\td b\tr\td c\td d\end{mtable}}
  \text{\spc\spc\spc and \spc\spc\spc}
M \spc=\spc\bal{\begin{mtable}{rr}\tr\td A\td B\tr\td C\td D\end{mtable}}
\spc\eos \]
If \(\lambda : \hsp F^2 \rightarrow F^2\) and \(\Lambda : \hsp F^{2n}
\rightarrow F^{2n}\) denote the linear endomorphisms corresponding
(relative to standard coordinates) to $m$ and $M$\aoc respectively,
then to what linear endomorphism that may be constructed from
$\lambda$ and $\Lambda$ may one relate the $4n \times 4n$ (Kronecker
product) matrix
\[ \bal{\begin{mtable}{rrrr}
    \tr\td aA \td bA \td aB \td bB
    \tr\td cA \td dA \td cB \td dB
    \tr\td aC \td bC \td aD \td bD
    \tr\td cC \td dC \td cD \td dD
   \end{mtable}} \spc\eoq\]
Explain your answer.
\end{enumerate}
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