<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%=========================================================================%
% initialize TeX
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\magnification=\magstep1
\baselineskip=14pt
\hsize = 6.5truein
\hoffset = 0.0truein
\vsize = 9.1truein
\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 (use \practice or \exam to initialize)
%=========================================================================%

\newcount\probcount
\probcount=0

\def\practice{
\def\prob##1\par
{\medskip\global\advance\probcount by 1\noindent
{\bf[\number\probcount]}~##1}}

\def\exam{
\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}}
% set up name, id, school fields for exam
\vskip 0.1in 
{\bf \noindent Name: \hrulefill \hbox{}}
\vskip 0.1in 
{\bf \noindent ID: \hrulefill ~ School: \hrulefill \hbox{}}
\vskip 0.2in
}

% set up those boxes to put the scores in

\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}}}

\def\fiveboxes#1{
\centerline{
\vbox{\offinterlineskip\halign{
 \strut\vrule\hfil##\hfil\vrule
 &amp;&amp;\hfil##\hfil\vrule\cr
 \noalign{\hrule}
 \boxA 1{#1} &amp;
 \boxA 2{#1} &amp;
 \boxA 3{#1} &amp;
 \boxA 4{#1} &amp;
 \boxA 5{#1} &amp;
 \boxT \cr
 \noalign{\hrule}
 \boxB &amp; \boxB &amp; \boxB &amp; \boxB &amp; \boxB &amp; \boxB  \cr
 \noalign{\hrule}
}}}}

\def\eightboxes#1{
\centerline{
\vbox{\offinterlineskip\halign{
 \strut\vrule\hfil##\hfil\vrule
 &amp;&amp;\hfil##\hfil\vrule\cr
 \multispan4{\hrulefill}\cr
 \boxA 1{#1} &amp;
 \boxA 2{#1} &amp;
 \boxA 3{#1} &amp;
 \boxA 4{#1} \cr
 \multispan4{\hrulefill}\cr
 \boxB &amp; \boxB &amp; \boxB &amp; \boxB \cr
 \noalign{\hrule}
 \boxA 5{#1} &amp;
 \boxA 6{#1} &amp;
 \boxA 7{#1} &amp;
 \boxA 8{#1} &amp;
 \boxT \cr
 \noalign{\hrule}
 \boxB &amp; \boxB &amp; \boxB &amp; \boxB &amp; \boxB\cr
 \noalign{\hrule}
}}}}

\def\tenboxes#1{
\centerline{
\vbox{\offinterlineskip\halign{
 \strut\vrule\hfil##\hfil\vrule
 &amp;&amp;\hfil##\hfil\vrule\cr
 \multispan5{\hrulefill}\cr
 \boxA 1{#1} &amp;
 \boxA 2{#1} &amp;
 \boxA 3{#1} &amp;
 \boxA 4{#1} &amp;
 \boxA 5{#1} \cr
 \multispan5{\hrulefill}\cr
 \boxB &amp; \boxB &amp; \boxB &amp; \boxB &amp; \boxB \cr
 \noalign{\hrule}
 \boxA 6{#1} &amp;
 \boxA 7{#1} &amp;
 \boxA 8{#1} &amp;
 \boxA 9{#1} &amp;
 \boxA{10}{#1} &amp;
 \boxT \cr
 \noalign{\hrule}
 \boxB &amp; \boxB &amp; \boxB &amp; \boxB &amp; \boxB &amp; \boxB\cr
 \noalign{\hrule}
}}}}

%==========================================================================
% 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\CC{\enma{\mathbb{C}}}
\def\QQ{\enma{\mathbb{Q}}}
\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\XX{\enma{\mathbf{X}}}
\def\KK{\enma{\mathbf{K}}}
\def\PP{\enma{\mathbf{P}}}
\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 Final Exam}
\smallskip
\centerline{Modern Algebra, Dave Bayer, May 12, 1999}
\medskip

%\practice
\exam
\eightboxes{5}

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 $\XX=\set{a_1,\ldots,a_n}\subset\CC$ be a finite set of points, where $\CC$
is the complex numbers. Define
$I\subset\CC[x]$ to be the set of all polynomials $f(x)$ that
vanish on every point of $\XX$. That is,
$$I \quad = \quad \setdef{f(x)\in \CC[x]}{f(a_i)=0 \;\;\hbox{for
every point}\;\; a_i\in \XX}.$$
\item{\bf(a)} Prove that $I$ is an ideal.
\item{\bf(b)} Give a set of generators for $I$.

\prob
What is the minimal polynomial of $\alpha=\sqrt{3}+\sqrt{5}$ over
$\QQ$?

\prob
Let $f(x,y,z) = x^3yz + xy^3z + xyz^3$. Express $f(x)$
as a polynomial $g(\sigma_1,\sigma_2,\sigma_3)$ where
$\sigma_1$, $\sigma_2$, $\sigma_3$ are the elementary symmetric
functions $$\sigma_1 = x+y+z,\quad \sigma_2 = xy+xz+yz,\quad \sigma_3 =
xyz.$$ 

\prob
Prove the primitive element theorem: Let $K$ be a finite
extension of a field $F$ of characteristic zero. There is an element
$\gamma\in K$ such that $K=F(\gamma)$.

\prob
Prove the following theorem about Kummer extensions: Let
$F$ be a subfield of $\CC$ which contains the $p$th root of unity $\zeta$
for a prime $p$, and let $K/F$ be a Galois extension of degree $p$. Then
$K$ is obtained by adjoining a $p$th root to $F$.

\noindent
Recall that the discriminant of $f(x)=x^3+px+q$ is $D=-4p^3-27q^2.$
\medskip

\prob
Let $f(x) = x^3+2$.
\item{\bf(a)} What is the degree of the splitting field $K$ of $f$ over
$\QQ$?
\item{\bf(b)} What is the Galois group $G=G(K/\QQ)$ of $f$?
\item{\bf(c)} List the subfields $L$ of $K$, and the corresponding
subgroups $H=G(K/L)$ of $G$.

\prob
Let $f(x) = x^3+x-2$.
\item{\bf(a)} What is the degree of the splitting field $K$ of $f$ over
$\QQ$?
\item{\bf(b)} What is the Galois group $G=G(K/\QQ)$ of $f$?
\item{\bf(c)} List the subfields $L$ of $K$, and the corresponding
subgroups $H=G(K/L)$ of $G$.

\prob
Let $f(x) = x^5-1$.
\item{\bf(a)} What is the degree of the splitting field $K$ of $f$ over
$\QQ$?
\item{\bf(b)} What is the Galois group $G=G(K/\QQ)$ of $f$?
\item{\bf(c)} List the subfields $L$ of $K$, and the corresponding
subgroups $H=G(K/L)$ of $G$.

\bye</pre></body></html>