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\title{Ph.D. Preliminary Examination in Algebra}
\date{June 4, 1999}
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\begin{document}
\begin{center}\LARGE\bf
Ph.D. Preliminary Examination in Algebra
\end{center}
\begin{center}
\large\bf
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
\[m \ =\ \left(\begin{array}{rr}
 a &  b\\
 c &  d\\
\end{array}
\right)
  \mbox{\ \ \  and \ \ \ }
M \ =\ \left(\begin{array}{rr}
 A &  B\\
 C &  D\\
\end{array}
\right)
\ \ \ . \]
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}