\input amstex
\documentstyle{amsppt}
\magnification=\magstep1
\pagewidth{6.5truein}
\pageheight{9truein}
\define\({\left(}
\define\){\right)}
\def\s#1#2{\medbreak
  \noindent{\bf#1.\enspace}{#2}\par
  \ifdim\lastskip<\medskipamount\removelastskip\penalty55\medskip\fi}
\def\t#1#2{\medbreak
  \centerline{{\bf#1.\enspace}{#2}}\par
  \ifdim\lastskip<\medskipamount\removelastskip\penalty55\medskip\fi}
\def\R{{\Bbb R}}
\nopagenumbers
\topmatter
\title
\bf \bf Ph.D. Comprehensive Examination\\
Abstract Algebra Section
\endtitle
\author
\bf August 1996
\endauthor
\endtopmatter
\document




\t{Part I}{Do three (3) of these problems.}

\s{I.1} {Let $(G,\cdot)$ be a group with binary operation $\cdot$, and let $a$
be a fixed element of $G$. Define a binary oparation $*$ on $G$ by setting $x
* y = x\cdot a \cdot y$, for $x$, $y \in G$. Show that $(G,*)$ is a group and
that it is isomorphic to $(G,\cdot)$.}
 
\s{I.2} {Let $R = \Bbb Z[x]$ be the ring of polynomials with coefficients in
$\Bbb Z$, the ring of integers; and let $p \in \Bbb Z$ be a prime number. Show
that $pR$, the ideal generated by $p$, is a prime ideal of $R$. Is $pR$
maximal? If so, explain why. If not, find the generators of a maximal ideal
which contains $pR$.}

\s{I.3}{\roster
 \runinitem"a)" Prove that a finite field must be of order $p^n$
for some prime $p$ and some positive integer $n$
 \item"b)" Show that for any such $p$ and any such $n$ there exists a field of
order $p^n$.
\endroster}

\s{I.4} {Let $P_n$ denote the set of real polynomials of degree $\leq n$, and
let $T:P_n \to P_n$ be defined by $T(p(t)) = (t-1)p'(t) + p(1)$.
\roster
 \item"a)" Prove that $T$ is linear.
 \item"b)" Find the matrix of $T$ with respect to the basis
$\{1,t,t^2,\dots,t^n\}$ for $P_n$.
 \item"c)" Find a basis for $P_5$ with respect to which the matrix of $T$ is
diagonal.
\endroster}

\vglue\baselineskip


\t{Part II}{Do two (2) of these problems.}

\s{II.1}{Let $G$ be a group and let $H$ be a subgroup of some finite index $n$
in $G$.
 \roster
 \item"a)" Show that $H$ contains a normal subgroup of $G$ whose index divides
$n!$. (Hint: Consider the action of $G$ on $G/H$ by right multiplication.)
 \item"b)" Show that $K = \bigcap_{a\in G} a Ha^{-1}$ is a normal subgroup of
$G$ and that any other normal subgroup of $G$ which is contained in $H$ is
contained in $K$.
 \item"c)" Show that if $K$, as defined above, consists of only of the identity
element, $e$, then $G$ can be embedded in a permutation group of order $n!$.
\endroster}

\s{II.2}
{Let $V$ be  a vector space of dimension $n$ over a field $K$. We call a linear
transformation $T:V\to V$ nilpotent if there exists an integer $N$ such that
$T^N = 0$, the zero map. Let $N$ be the smallest such integer.
 \roster 
 \item"a)" Show that if $T$ is nilpotent, then $T^k(V) \subset T^{k-1}(V)$ and
$\dim T^k(V) < \dim T^{k-1}(V)$ for every integer $k$, $1\leq k \leq N$.
 \item"b)" Show that if $T$ is nilpotent, then $T^n = 0$ (i.e\., $N \leq n$).
 \item"c)" Prove that if $T$ is nilpotent, then $(T-I)$ is invertible, where
$I$ is the identity map.
\endroster}

\s{II.3}
{Let $p$ be a prime number, and let $K$ be a field of characteristic $q \ne p$
which contains all $p^{\roman {th}}$ roots of unity. Let $a$ be an element of
$K$ which is not a $p^{\roman{th}}$ power in $K$, and let $\alpha$ be a
$p^{\roman{th}}$ root of $a$ in an algebraic closure $\overline K$ of $K$. 
\roster
\item"a)" Determine $[K(\alpha):K]$.
\item"b)" Determine the Galois group $\roman{Gal}(K(\alpha);K)$ of $K(\alpha)$
over $K$.
\item"c)" Determine all extensions of the field $K$ contained in the field
$K(\alpha)$.
\endroster}

\enddocument