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\raggedleft{August, 1998}
\bigskip

\noindent{\bf Comprehensive Examination \hfill 
        Department of Mathematics}
\vskip .5in

\centering{{\bf ALGEBRA}\\
\vskip .7in PART I: Do three of the following problems.}
\bigskip

\begin{enumerate}

\item Let $G$ be a group. 
        \begin{enumerate}
        \item   Show that $G$ is finite iff $G$ has only
                finitely many distinct subgroups.
        \item Show that $G$ has exactly 3 distinct subgroups 
                iff $G$ is cyclic of order $p^2$ for some 
                prime $p$.
        \end{enumerate}

\item Let $A$ be a real $r\times r$-matrix satisfying $A^n=I$ 
        for some $n>0$. Prove: $\det A=(-1)^m$, where $m$ is 
        the multiplicity of $-1$ as root of the characteristic 
        polynomial of $A$.

\item Let $R$ be a ring and let $N$ be an ideal of $R$ such 
        that every element of $x\in N$ is nilpotent, that is, 
        $x^t=0$ for some $t$. Show that, under the canonical map 
        $R\rightarrow R/N$, the group of units $\U(R)$ of 
        $R$ maps {\it onto} the group of units $\U(R/N)$ of $R/N$. 
        (Recall that a {\it unit} of a ring $R$ is an invertible 
        element of $R$.)
\item Let $F\supseteq K$ be an algebraic extension of fields and 
        let $R$ be a subring of $F$ with $R\supseteq K$. Show that 
        $R$ is a field.


\end{enumerate}


\newpage

\centering{PART II: Do two of the following problems.}
\bigskip

\begin{enumerate}

\item Let $G$ be a group of order $pqr$ with distinct primes $p$, 
        $q$, and $r$. Show that $G$ is not simple.

\item Let $R=K[x,y]$ be the ring of polynomials in two
        variables $x$ and $y$ with coefficients in the field $K$, 
        and let $f(x,y)\in R$.
        \begin{enumerate} 
        \item Show that the principal ideal of $R$ that is generated 
                by $f(x,y)$ is prime if and only if the polynomial 
                $f(x,y)$ is irreducible.
        \item  Show that the ideal of $R$ that is generated by $x$ and 
                $f(x,y)$ is maximal if and only if the polynomial $f(0,y)$ 
                is irreducible in $K[y]$.
      \end{enumerate}

\item Let $F$ be the splitting field of $x^{6}-3$ over ${\bf Q}$.
        \begin{enumerate} 
        \item Show that  $[F:{\bf Q}]=12$.
        \item Let $G=Gal(F/{\bf Q})$. Show that there exist a normal 
                subgroup $H$ of $G$ of order $6$ and a subgroup $K$ of 
                $G$ of order $2$ such that $G$ is a semidirect product 
                of $H$ and $K$.
        \item Determine whether the subgroup $H$ of Part (b) is 
                abelian.
        \end{enumerate}
      
\end{enumerate}



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