%=========================================================================%
% initialize TeX
%=========================================================================%

\magnification=\magstep1
\baselineskip=14pt
\hsize = 6.5truein
\hoffset = 0.0truein
\vsize = 9.5truein
\voffset = -0.25truein
\emergencystretch = 0.05\hsize

\newif\ifblackboardbold

% comment out the following line if AMS msbm fonts aren't available
\blackboardboldtrue

%=========================================================================%
% select fonts
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\font\titlefont=cmbx12 scaled\magstephalf
\font\sectionfont=cmbx12
\font\netfont=cmtt9

% Establish AMS blackboard bold fonts without using amssym.def, amssym.tex

\newfam\bboldfam
\ifblackboardbold
\font\tenbbold=msbm10
\font\sevenbbold=msbm7
\font\fivebbold=msbm5
\textfont\bboldfam=\tenbbold
\scriptfont\bboldfam=\sevenbbold
\scriptscriptfont\bboldfam=\fivebbold
\def\bbold{\fam\bboldfam\tenbbold}
\else
\def\bbold{\bf}
\fi

%=========================================================================%
% exam macros (these are different from corresponding practice macros)
%=========================================================================%

\newcount\probcount
\probcount=0

\def\prob#1\par
{\global\advance\probcount by 1\noindent {\bf[\number\probcount]}~#1
\vfill\eject
\noindent{\bf Problem: \hbox to 30pt{\hrulefill}}
\vfill\eject
}

\footline={\tenbf Page \folio\hfil Continued on page: \hbox to
30pt{\hrulefill}}

%==========================================================================
% macros
%==========================================================================

% surround with $ $ if not already in math mode
\def\enma#1{{\ifmmode#1\else$#1$\fi}}

\def\mathbb#1{{\bbold #1}}
\def\mathbf#1{{\bf #1}}

% blackboard bold symbols
\def\FF{\enma{\mathbb{F}}}
\def\NN{\enma{\mathbb{N}}}
\def\RR{\enma{\mathbb{R}}}
\def\ZZ{\enma{\mathbb{Z}}}
\def\bbPP{\enma{\mathbb{P}}}

% caligraphic symbols
\def\cAA{\enma{\cal A}}
\def\cBB{\enma{\cal B}}

% bold symbols
\def\aa{\enma{\mathbf{a}}}
\def\bb{\enma{\mathbf{b}}}
\def\cc{\enma{\mathbf{c}}}
\def\ee{\enma{\mathbf{e}}}
\def\mm{\enma{\mathbf{m}}}
\def\pp{\enma{\mathbf{p}}}
\def\vv{\enma{\mathbf{v}}}
\def\ww{\enma{\mathbf{w}}}
\def\xx{\enma{\mathbf{x}}}
\def\yy{\enma{\mathbf{y}}}
\def\CC{\enma{\mathbf{C}}}
\def\XX{\enma{\mathbf{X}}}
\def\KK{\enma{\mathbf{K}}}
\def\PP{\enma{\mathbf{P}}}
\def\QQ{\enma{\mathbf{Q}}}
\def\LL{\enma{\mathbf{L}}}
\def\II{\enma{\mathbf{I}}}
\def\zero{\enma{\mathbf{0}}}
\def\blambda{\enma{\mathbf{\lambda}}}

% misc

\def\abs#1{\enma{\left| #1 \right|}}
\def\set#1{\enma{\{#1\}}}
\def\setdef#1#2{\enma{\{\;#1\;\,|\allowbreak
  \;\,#2\;\}}}
\def\idealdef#1#2{\enma{\langle\;#1\;\,
  |\;\,#2\;\rangle}}

%=========================================================================%
% exam
%=========================================================================%

\centerline{\titlefont Second Midterm Exam}
\smallskip
\centerline{Modern Algebra, Dave Bayer, March 31, 1999}
\medskip

\vskip 0.1in 
{\bf \noindent Name: \hrulefill \hbox{}}

\vskip 0.1in 
{\bf \noindent ID: \hrulefill ~ School: \hrulefill \hbox{}}

\vskip 0.2in 

\def\boxAsub#1{\vbox{\hbox{#1}}}
\def\boxA#1#2{[{\bf #1}] (#2 pts)}
\def\boxT{\bf TOTAL}
\def\boxB{\vbox to 25pt{\hbox{\kern 56pt}}}

\centerline{
\vbox{\offinterlineskip\halign{
 \strut\vrule\hfil#\hfil\vrule
 &&\hfil#\hfil\vrule\cr
 \noalign{\hrule}
 \boxA 16 &
 \boxA 26 &
 \boxA 36 &
 \boxA 46 &
 \boxA 56 &
 \boxT \cr
 \noalign{\hrule}
 \boxB & \boxB & \boxB & \boxB & \boxB & \boxB  \cr
 \noalign{\hrule}
}}}

{\tenrm
Each problem is worth 6 points, for a total of 30 points. Please work only one
problem per page, and label all continuations in the spaces provided. Extra pages
are available. Check your work, where possible.}

\bigskip

\prob
Let $G$ be the abelian group $G=\idealdef{a,b,c}{a^7bc^2 =
ab^4c^5 = abc^2 = 1}$. Express $G$ as a product of free and cyclic groups.


\prob
What is the minimal polynomial of $\alpha=\sqrt{-1}+\sqrt{2}$ over
$\QQ$?

\prob
Consider the module $M=F[x]/(x^3+3x^2+3x+1)$ over the ring $R=F[x]$ for a field
$F$.
\item{\bf(a)} What is the dimension of $M$ as an $F$-vector space?
\item{\bf(b)} Find a basis for $M$ as an $F$-vector space, for which the matrix
representing multiplication by $x$ is in Jordan canonical form. Give this
matrix.

\prob
Prove the {\it Eisenstein criterion\/} for irreducibility: Let $f(x)=a_n
x^n + \ldots + a_1 x + a_0 \in \ZZ[x]$, and let $p$ be a prime. If $p$
doesn't divide $a_n$, $p$ does divide $a_{n-1}, \ldots, a_0$, but $p^2$
doesn't divide $a_0$, then $f(x)$ is irreducible as a polynomial in
$\QQ[x]$.

\prob
Let $R$ be a principal ideal domain, and let
$$I_1 \quad\subset\quad I_2 \quad\subset\quad I_3
 \quad\subset\quad \cdots \quad\subset\quad I_n \quad\subset\quad \cdots$$
be an infinite ascending chain of ideals in $R$. Show that this chain {\it
stabilizes\/}, i.e.
$$I_N \quad=\quad I_{N+1} \quad=\quad I_{N+2} \quad=\quad \cdots$$
for some $N$.

\bye