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\title{Ph.D. Preliminary Examination in Algebra}
\date{June 4, 1999}
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\begin{document}
\begin{center}\LARGE\bfseries{}
Ph.D. Preliminary Examination in Algebra
\end{center}
\begin{center}
\large\bfseries{}
June 4, 1999
\end{center}

\medskip
\begin{enumerate}

\item  Let \(A\) be an \(n \times{} n\) matrix with entries in the field %
\(\mbox{\textbf{C}}\) 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 \(2\)\@. \ 

\item  A nested sequence of finite groups \,\(G, \, H, \, K\)\, with %
       \(H\) a normal subgroup of \(G\) and \(K\) a normal subgroup of \(H\) %
       such that \(K\) is not a normal subgroup of \(G\)\@. \ 
\end{enumerate} %


\item  Let \(p\) be the polynomial %
\,\(p(t)  \, = \,  t^{5}  +  t^{2} +  1\)\, %
regarded as an element of the ring \,\(A \, = \, \mbox{\textbf{F}}_{2} [t]\)\, %
of polynomials with coefficients in the field \,\(\mbox{\textbf{F}}_{2}\)\, 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 \(N\)\@, \, and let \(n\) be a %
positive integer that divides \(N\)\@. \   Do \textbf{one} of the following: %
\begin{enumerate}

\item  Prove that if \(G\) is abelian, then \(G\) contains a subgroup of %
       order \(n\)\@. \ 

\item  Find an example of \,\(G, \, N, \, n\)\, as above where \(G\) %
       has no subgroup of order \(n\)\@. \ 
\end{enumerate} %


\item  Show that every group of order \(30\) contains a normal cyclic %
subgroup of order \(15\)\@. \  %


\item  Let \(F\) be the field \,\(\mbox{\textbf{Q}}(i)\)\, where %
\,\( i \, = \, \sqrt{-1} \in{} \mbox{\textbf{C}}\)\,,\  and let \(E\) be the splitting %
field over \(F\) of the polynomial %
\,\( f(t) \, = \, t^{4} - 5 \)\,. \    Find: %
\begin{enumerate}

\item  the extension degree \,\([E:F]\)\@. \ 

\item  the group \,\(\mbox{Aut}_{F}(E)\)\, of all automorphisms of %
       \(E\) that fix \(F\)\@. \ 
\end{enumerate} %


\item  Let \,\(\mbox{\textbf{F}}_{2}\)\, be the field of \(2\) elements, and let %
\(R\) be the commutative ring %
\[ R \  \ = \  \  \mbox{\textbf{F}}_{2} [t]/t^{3} \mbox{\textbf{F}}_{2} [t] \, . \] %
\begin{enumerate}

\item  How many elements does \(R\) contain? \ 

\item  What is the characteristic of \(R\)\@? \ 

\item  Find all ring homomorphisms %
           \,\( R \, \rightarrow{} \, R \)\,.
\end{enumerate} %


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

