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\title{M.A. Algebra Comprehensive Exam}
\author{}
\date{\today}
\begin{document}
\maketitle
{\em Do {\bf two} problems from each of the three parts. Please give complete proofs. If you try three problems in one ot the three parts, please indicate which two problems you wish to be graded.}\\
\begin{center}Part I
\end{center}
%\noindent{\bf1.} (a) Prove the theorem of Cauchy, that if $G$ is a
%finite group and $p$ is a prime divisor of the order of $G$, then
%$G$ contains an element of order $p$.
%(b) Give a specific example to show that the statement of Cauchy's
%theorem is false if $p$ is not a prime number.\\
\noindent{{\bf1.} Determine whether each of the following
statements is true or false. If true, prove it. If false, describe
a counterexample.
(a) If $G$ is a group and $N$ is a normal subgroup of $G$ such that both $N$ and $G/N$ are abelian, then $G$ is abelian.
(b) If $H$ is a finite subgroup of a group $G$ and $N$ is a normal subgroup of $G$ such that the index $|G:N|$ is finite and relatively prime to $|H|$, then $H \le N$.\\
\noindent{\bf2.} (a) If $G$ is a group and $H$ is a subgroup of
$G$ of index $|G:H| = n < \infty$, prove that there is a
homomorphism $G \rightarrow S_n$ whose kernel is contained in $H$.
(b) If $G$ is a non-abelian simple group, prove that $G$ has no
subgroup of index $3$.\\
\noindent{\bf3.} (a) If $G$ is a cyclic group, prove that the
automorphism group of $G$ is abelian.
(b) If $G$ is a group whose automorphism group is cyclic, prove
that $G$ is abelian.
({\em Hint: Consider the group of inner automorphisms.})\
\begin{center}Part II
\end{center}
\noindent{\bf4.} (a) If $F$ is a field, show that $F[x]$ is a
principal ideal domain (P.I.D.).
(b) If $D$ is an integral domain such that $D[x]$ is a P.I.D.,
show that $D$ is a field.\\
\noindent{\bf5.} An ideal $P$ of a commutative ring $R$ (with $1$)
is called a {\em prime} ideal if, whenever $a, b \in R$ such that
$ab \in P$, then either $a \in P$ or $b \in P$.
(a) Prove that any maximal ideal of $R$ is a prime ideal.
(b) Give an example of a commutative ring $R$ and an ideal $P$ of
$R$ such that $P$ is prime but not maximal.\\
\noindent{\bf6.} (a) Show that each of the following polynomials
is irreducible over the field indicated:
\hspace{1cm}(i) $x^3 -x + 1 \in \Z_3[x]$, where $\Z_3$ is the ring
of integers modulo $3$
\hspace{1cm}(ii) $x^4 + x^3 + x^2 + x + 1 = \frac{x^5 - 1}{x - 1}
\in \Q[x]$, where $\Q$ is the field of rational numbers
(b) Describe the construction of a field having $27$ elements.
%\noindent{\bf6.} Prove that if an integral domain $D$ (with $1$)
%has only finitely many distinct ideals, then it is a field.\\
\begin{center}Part III
\end{center}
\noindent{\bf7.} Let $V$ be finite-dimensional vector space over
the reals, $\R$, and let $T$ be a linear operator on $V$. Suppose
that $T$ commutes with every diagonalizable linear operator on
$V$. Prove that $T$ is a scalar multiple of the identity
operator. \\
\hspace{11cm} Continued on next page \vspace{2cm}
\noindent{\bf8.} (a) Let $V$ and $W$ be vector spaces and let $T$
be a linear operator from $V$ into $W$. Suppose that $V$ is
finite-dimensional. Prove
$\mbox{rank}(T)+\mbox{nullity}(T)=\mbox{dim}(V)$.
(b) Let $S$ be the linear operator defined on the space of
$3\times 3$ real matrices given by,
\[ S(A)=A-A^t,\]
where $A^t$ denotes the transpose of the matrix $A$. Determine
the rank of $S$.\\
\noindent{\bf9.} Suppose $T:V\to W$ is a linear map between finite
dimensional real vector spaces.
(a) Give the definition of the matrix $A_T$ of $T$ with respect to
bases of $V$ and $W$.
(b) Prove that $T$ is invertible if and only if $A_T$ is
invertible.
(c) If $V=W=\{\mbox{real polynomials of degree} \le n\}$ and
$T=\frac{d}{dx}$ then find the matrix of $T$ with respect to your
favorite basis.
\end{document}