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\begin{document}
\bc
\large\bf DEPARTMENT OF MATHEMATICS \& STATISTICS\\
\Large\bf Preliminary Ph.D. Examination in Algebra\\
\large\bf September 2, 1993
\ec

\bigskip
\bhp{1.}{
Determine the number of 3 $\times$ 3 invertible matrices in a finite field
having  $q$  elements.
}\ehp

\bigskip
\bhp{2.}{
Can a line segment with length equal to the positive real fifth
root of  $2$  be constructed (given unit length) in a finite
number of steps using straightedge and compass?  Explain.
}\ehp

\bigskip
\bhp{3.}{
Let  A  be a commutative ring (with multiplicative identity).}\vspace{8bp}\\
\hwpart{(a)}{Let  $M$  be an $A$-module.  Let  $\,R = \mbox{End}(M)\,$
be the set of endomorphisms of  $M$  (i.e., the set of  $A$-module
homomorphisms  $\,M \rightarrow M\,$).  Define operations on  $R$  that
make  $R$  an $A$-algebra (i.e., a ring with compatible $A$-module
structure).}\\
\hwpart{(b)}{Is the set of ring endomorphisms (as opposed to A-module 
endomorphisms) of the ring  $A$  a ring?}\\
\hwpart{(c)}{When  $A = \Z$ ,  the ring of integers,  find the endomorphism
ring of the  $\Z$-module   $\,\Z \oplus \Z/4\Z$\@.
}\ehp

\bigskip
\bhp{4.}{Show that any group of order $20$ has a non-trivial proper
normal subgroup.
}\ehp

\bigskip
\bhp{5.}{Prove that a finitely-generated torsion-free module over a
principal ideal domain is necessarily free.
}\ehp

\bigskip
\bhp{6.}{Determine the isomorphism class of each of the Sylow
subgroups of the alternating group  $A_5$\@, the group of ``even''
permutations of a set of cardinality $5$\@.
}\ehp

\bigskip
\bhp{7.}{
Let  $\zeta$  be a primitive $7^{\mbox{th}}$ root of unity in the
field of complex numbers,  let  $K = \Q(\zeta)$\@,  and  $H = \Q(\alpha)$\@,
\ where  $\,\alpha = cos(2\pi/7)\,$ and $\Q$  denotes the field of rational
numbers.}\vspace{8bp}\\
\hwpart{(a)}{Show that  $H = K \cap \R$\@,  where  $\R$  is the field of
real numbers.}\\
\hwpart{(b)}{Prove that  $K$  and  $H$  are both normal extensions
of  $\Q$\@.}\\
\hwpart{(c)}{Determine the Galois groups  $\Gal(K:\Q)$\@,  $\Gal(H:\Q)$\@,
and $\Gal(K:H)$\@.}\ehp

\bigskip
\bhp{8.}{
For  $p$  a prime the ring of $p$-adic integers  $\Z_p$  is defined
to be the inverse limit of the unique ring homomorphisms
\[
  \ldots \ \rightarrow \ \Z/p^n\Z \ \rightarrow \ \ldots \ \rightarrow
  \ \Z/p^2\Z \ \rightarrow \ \Z/p\Z \ .
\]
Let  $\pi$  denote the canonical ring homomorphism
$\,\Z_p \rightarrow \Z/p\Z$\@.
}\vspace{8bp}\\
\hwpart{(a)}{Show that an element of  $\Z_p$  is invertible in  $\Z_p$
if and only if its image under  $\pi$  is non-zero.}\\
\hwpart{(b)}{Show that the kernel of  $\pi$  is a maximal ideal of
$\Z_p$\@.}\\
\hwpart{(c)}{Show that any proper ideal in  $\Z_p$  is contained in
the kernel of  $\pi$\@.  (Hence,  $\,\mbox{ker}(\pi)\,$  is the
only maximal ideal.)}\\
\hwpart{(d)}{Show that the kernel of  $\pi$  is a principal ideal.}\\
\hwpart{(e)}{Give another (non-isomorphic) example of a ring having a
unique maximal ideal that is principal.}
\ehp

\end{document}