%=========================================================================% % 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