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

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

\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

%=========================================================================%
% problem macro (this is different from exam problem macro)
%=========================================================================%

\newcount\probcount
\probcount=0

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

%==========================================================================
% 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 Practice Problems for Final Exam}
\smallskip
\centerline{Modern Algebra, Dave Bayer, May 3, 1999}
\medskip

\bigskip

Our final will be held on Wednesday, May 12, 1:10pm -- 4:00pm, in our regular
classroom. It will consist of 8 questions worth 40 points in all. Two questions
will be review from previous exams, and the remaining six questions
will test material since the last exam.

The following two problems constitute the review topics from previous exams.

\bigskip

\prob (compare with midterm 1, problem 5)
Let $\XX\subset\RR^2$ be a finite set of points.
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,
$$I \quad = \quad \setdef{f(x,y)\in \RR[x,y]}{f(a,b)=0 \;\;\hbox{for
every point}\;\; (a,b)\in X}.$$
Prove that $I$ is an ideal.

\bigskip

\prob (midterm 2, problem 2)
What is the minimal polynomial of $\alpha=\sqrt{-1}+\sqrt{2}$ over
$\QQ$? (Note that we now have another way to compute this; see Artin, p. 554,
discussion after proof of Proposition 14.4.4.)

\bigskip

The following problems are taken from last year's review questions for the
final.

\bigskip

\prob
Let $f(x,y,z) = x^2y + x^2z + xy^2 + y^2z + xz^2 + yz^2$. 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 $$\sigma_1 = x+y+z,\quad \sigma_2 = xy+xz+yz,\quad \sigma_3 =
xyz.$$ 

\bigskip

\prob
Let $f(x) = x^2+ax+b$ have roots $\alpha_1$ and $\alpha_2$, where
$$\alpha_1 + \alpha_2 = c, \qquad \alpha_1^2+\alpha_2^2 = d.$$
Express $a$ and $b$ in terms of $c$ and $d$.

\bigskip

Recall that the discriminant of $f(x)=x^2+bx+c$ is $D=b^2-4c$, and that
the discriminant of $f(x)=x^3+px+q$ is $D=-4p^3-27q^2.$

\bigskip

\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 $G$.

\bigskip

\prob
Repeat for $f(x) = x^3-3x+1$.

\bigskip

\prob
Repeat for $f(x) = x^4-3x^2+2$.

\bigskip

\prob
Repeat for $f(x) = x^4-5x^2+6$.

\bigskip

\prob
Prove the primitive element theorem (14.4.1, p. 552): Let $K$ be a finite
extension of a field $F$ of characteristic zero. There is an element
$\gamma\in K$ such that $K=F(\gamma)$.

\bigskip

\prob
Prove the following theorem about Kummer extensions (14.7.4, p. 566): 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 $F$.

\bigskip

The following problems are taken from last year's final.

\bigskip

\prob
Prove that $\alpha=e^{2\pi i /11} + 3$ is not constructible.

\bigskip

\prob 
What is the minimal polynomial of $\alpha=\sqrt{2}+\sqrt{3}$ over
$\QQ$?

\bigskip

\prob
Let $f(x,y,z) = x^3 + y^3 + z^3$. 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 $$\sigma_1 = x+y+z,\quad \sigma_2 = xy+xz+yz,\quad \sigma_3 =
xyz.$$ 

\bigskip

Recall that the discriminant of $f(x)=x^2+bx+c$ is $D=b^2-4c$, and that
the discriminant of $f(x)=x^3+px+q$ is $D=-4p^3-27q^2.$

\bigskip

\prob
Let $f(x) = x^3-12$.
\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 $G$.

\bigskip

\prob
Prove the primitive element theorem (14.4.1, p. 552): Let $K$ be a finite
extension of a field $F$ of characteristic zero. There is an element
$\gamma\in K$ such that $K=F(\gamma)$.
\bigskip



\bye