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\topmatter
\title
\bf \bf Ph.D. Comprehensive Examination\\
Abstract Algebra Section 
\endtitle
\author
\bf January 1996
\endauthor
\endtopmatter
\document




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

\s{I.1} {\roster
 \runinitem "a)" Let $G$ be a group with a subgroup $H$ and a
normal subgroup $N$. Prove that $H\cap N$ is a normal
subgroup of $H$ and that $H/(H\cap N) \cong HN/N$.
 \item "b)" Use the result in part (a) to prove that a subgroup
$H$ of a solvable group is solvable. (Note: A group $G$
is solvable if there exists a subnormal series $\{1\} =
N_k \triangleleft N_{k-1} \triangleleft \cdots
\triangleleft N_2 \triangleleft N_1 \triangleleft N_0
\triangleleft G$, where, for each $i = 1,\dots,k$,
$N_{i-1}/N_i$ is abelian.)
\endroster}
 
\s{I.2} {Let $R$ be an Euclidean domain with unity.
 \roster
 \item "a)" Prove that $R$ is a principal ideal domain. 
 \endroster 
 For $r$ and $s$ in $R$,
let $\langle r, s \rangle  = \{rm+sn \,|\, m,\ n \in R\}$
 \roster
 \item "b)" Prove that $\langle r, s \rangle$ is an ideal of $R$.
 \item "c)" Suppose that $R= Q[x]$ (where $Q$ represents the field of rational
numbers), and that $r = x^2 - 3 x + 2$ and $s = x^3 - 9x^2 + 23 x - 15$. Then,
by parts (a) and (b), $\langle r, s \rangle = \langle p \rangle$, for some $p
\in Q[X]$. Find $p$. (Show all work and justify your answer.)
 \endroster}
 
\s{I.3} {
Let $F$ be a field. Then $F[x]$ is a commutative
ring with unity. (You may accept this without proof.)
 \roster
 \item "a)" Show that $F[x]$ is an integral domain but that it
cannot be a field.
 \endroster
Let $E$ be an extension field of $F$. For $\alpha \in
E$, let $\phi: F[x] \to E$ be a homomorphism which
fixes the elements of $F$ and which maps $x$ into
$\alpha$. 
 \roster
 \item "b)" Describe the kernel of $\phi$.
 \item "c)" Prove that if $\alpha$ is algebraic then the image of $\phi$ is a
subfield of $E$.
 \item "d)" For $F = Q$ (the rationals) and for $\alpha = \root
3 \of 2$, describe the image of $\phi$ as a subfield of
$R$ (the reals). (I.e., find a basis and a unique
representation for each element.)
 \endroster}

\s{I.4} { Let $V$ be a finite dimensional vector space with
an inner product $\langle \cdot, \cdot \rangle$. Fix a
non-zero vector $u$ in $V$ and define a mapping $T:V
\to V$ by $T(v) = \langle u, v \rangle u$. 
 \roster
 \item "a)" Show that $T$ is a linear transformation.
 \item "b)" Find the characteristic poynomial, eigenvalues,
minimal polynomial, and Jordan canonical form of $T$. 
 \item "c)" If, instead of $T$, we consider the mapping $F:V\to
V$ given by $F(v) = \langle u, v \rangle v$, explain
why the analogues of the questions asked in part b) are
not meaningful for $F$.
 \endroster}

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

\s{II.1}{\roster
 \runinitem "a)" Find all groups of order $325$.
 \item "b)" Find all groups of order $22$.
 \endroster}

\s{II.2}
{Let $A$ be an $m \times n$ matrix over a field $F$. 
\roster
\item "a)" Show that the rank of $A$ is equal to the smallest
integer $r$ such that $A$ can be factored as $A = BC$
for suitable matrices $B$ and $C$ of sizes $m \times r$
and $r \times n$ respectively. 
\item "b)" Use part (a) to deduce the familiar fact that ``row
rank = column rank''.
 \endroster}

\s{II.3}
{Let $F$ be a finite field with $q$ elements.
 \roster
 \item "a)" Prove that the product of the non-zero elements of
$F$ is $-1$. 
 \item "b)" Prove that if $q$ is even then every element of $F$ is a square; and
that if $q$ is odd, then the set of non-zero squares of $F$ is a subgroup of
index $2$ of the group of non-zero elements of $F$.
 \endroster}
\enddocument