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% initialize TeX
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\magnification=\magstephalf
\baselineskip=13pt
\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

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% 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

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\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
%=========================================================================%

\def\prob#1{\noindent {\bf [#1]}}

\def\endprob{
\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}}

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% exam
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\centerline{\titlefont First Midterm Exam}
\smallskip
\centerline{Modern Algebra, Dave Bayer, February 17, 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.}


\bigskip

\prob 1
Consider the ideal $I=(x^3,2x,4)\subset \ZZ[x]$.
\item{\bf(a)} List representatives
for the elements of the quotient ring $R=\ZZ[x]/I$, and describe the
multiplication rule in $R$ for these representatives.
\item{\bf(b)} Is $R$ an integral domain?

\endprob

\prob 2
Prove that if a ring $R$ has no ideals other than $(0)$ and $(1)$, then $R$
is a field.

\endprob

\prob 3
Prove that if an integral domain $R$ has only finitely many elements,
then $R$ is a field. Prove any lemmas that you use.

\endprob

\prob 4
Let $\FF_3$ be the finite field with 3 elements. 
Find a polynomial
$f(x)$ in the polynomial ring $\FF_3[x]$ such that the quotient ring
$\FF_3[x]/(f(x))$ is a field with 27 elements.

\endprob

\prob 5
Let $\XX\subset\RR^2$ be the union of the parabola $y=x^2$ and the point $(0,1)$.
Define
$I\subset\RR[x,y]$ to be the set of all polynomials $f(x,y)$ that
vanish on every point of $\XX$. That is,
$$I \quad = \quad \setdef{f(x,y)\in \RR[x,y]}{f(a,b)=0 \;\;\hbox{for
every point}\;\; (a,b)\in X}.$$
\item{\bf(a)} Prove that $I$ is an ideal.
\item{\bf(b)} Give a set of generators for $I$.
\bigskip

\endprob

\bye