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\begin{document}
\bc
\large\bf Department of Mathematics and Statistics\\
University at Albany\\
\Large\bf Preliminary Ph.D. Examination in Algebra\\
\large\bf June 16, 1995
\ec

\bigskip
\btp{[10]}{1.}{Let $\Q$ denote the field of rational numbers.
Prove {\bf or} disprove {\bf one} of the following assertions
concerning an arbitrary {\em symmetric} matrix $M$ with entries in $\Q$\@.
There exists an invertible matrix $P$ with entries in $\Q$ such that:}\\
\expart{(a)}{$PM\,^{t}\hspace{-2bp}P$ is diagonal.}\\
\expart{(b)}{$PMP^{-1}$ is diagonal.
}
\etp

\bigskip
\btp{[10]}{2.}{Let $k$ be a field.}\\
\expart{(a)}{Show that $\ (x+1)\ $ is a maximal ideal in the polynomial
             ring $\ k[x]$\@.}\\
\expart{(b)}{Show that $\ (x+1,\ y-2)\ $ is a maximal ideal of $\ k[x,y]$\@.}\\
\expart{(c)}{Let $A$ be the quotient ring \ \mbox{$k[x,y]/(x+1)$}\@,\ \ \ and
            let $\varphi$ be the $k[x,y]$-linear endomorphism of $A$ given by
             \[  \varphi(f) = (y-2)f \ . \]
             Show that the cokernel of $\varphi$ is a $1$-dimensional
             $k$-module.}
\etp

\bigskip
\btp{[10]}{3.}{Show that every automorphism of the symmetric group
  $S_3$ (the group of all permutations of a set with $3$ members) is an
inner automorphism.}
\etp

\bigskip
\btp{[10]}{4.}{Let $E$ be a (finite) Galois extension field of $F$ with
Galois group $G$\@; let $K$ be an intermediate field and $H$ the subgroup
of $G$ that fixes each member of $K$\@.  Show that the subgroup of $G$
consisting of all $\sigma$ in $G$ for which \mbox{$\sigma(K)=K$} is the
{\em normalizer} of $H$ in $G$\@, i.e., the largest subgroup $N$ of $G$
containing $H$ for which $H$ is a normal subgroup of $N$\@.}
\etp

\bigskip
\btp{[10]}{5.}{For $K$ any field \mbox{$\mbox{GL}(n, K)$} denotes the
group of invertible \mbox{$n \times n$} matrices in the field $K$\@, and
\mbox{$\mbox{SL}(n, K)$} denotes the group of such matrices of
determinant $1$\@.
Prove that
\mbox{$\mbox{GL}(n, K)$} is isomorphic to a semi-direct product of
\mbox{$\mbox{SL}(n, K)$} with \mbox{$\mbox{GL}(1, K)$}\@.
}
\etp

\bigskip
\btp{[10]}{6.}{Let $R$ be a commutative ring with a {\bf unique} maximal
ideal $M$\@. Show that if $e^2=e$\@, then $e=0$ or $e=1$\@.}
\etp

\bigskip
\btp{[10]}{7.}{For $I$ and $J$ ideals in a commutative ring $R$,
prove that the natural $R$-algebra homomorphism
\[  R/I  \otimes_{R}  R/J  \longrightarrow  R/(I+J)  \]
is an isomorphism of $R$-algebras.}
\etp

\bigskip
\btp{[10]}{8.}{Let  $K$  be the splitting field over the field $\Q$
of rational numbers of the polynomial
\[ t^4 + 4 t^2 + 2 \ . \]
Find the Galois group of $K$ over $\Q$\@.
}
\etp

\end{document}