\documentclass{article} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{enumerate} \usepackage{ifthen} \usepackage{graphicx} \setlength{\unitlength}{0.1in} \setlength{\oddsidemargin}{0.0in} \setlength{\evensidemargin}{0.0in} \setlength{\topmargin}{0.0in} \setlength{\headheight}{0.0in} \setlength{\headsep}{0.0in} \setlength{\topskip}{0.0in} \setlength{\textheight}{8.9in} \setlength{\textwidth}{6.5in} \newcommand{\comp}{ \,{\scriptstyle \stackrel{\circ}{}}\, } \newcommand{\nullset}{\mathrm{O}\!\!\!\!\big/\,} \newcommand{\divides}{\,\Big|\,} \newcommand{\myspace}{\mbox{\parbox[c][0.5in]{0.45in}{\hfill}}} \newcommand{\varspace}[2]{\mbox{\parbox[c][#1]{#2}{\hfill}}} \newcommand{\diagquotient}[2]{\displaystyle{{#1 \atop \;}\hspace*{-0.1in} \mbox{\put(0,0){\line(2,1){2}}} \hspace*{0.1in} {\; \atop #2}}} \begin{document} \pagestyle{empty} \noindent \parbox{2in}{\bf Math 3110} \hfill {\Large \bf Final Exam} \hfill \parbox{2in}{\bf \hfill December $7^{\mathrm{th}}$, 2015} \vspace{0.3in} \noindent {\large\bf Name:} \underline{\hskip 3.0 truein} \hfill {\bf Be sure to show your work!} \vspace{0.2in} \noindent {\large 1. (10 points)} Working in $\mathbb{Z}_{18}$. \begin{enumerate}[(a)] \item $I = (6)=\langle 6 \rangle = {\Big\{} \underline{\hspace*{2in}} {\Big\}}$ and $\diagquotient{\mathbb{Z}_{18}}{I} = {\Big\{} \underline{\hspace*{2in}} {\Big\}}$. \item For each element in $\diagquotient{\mathbb{Z}_{18}}{I}$, state whether that element is zero, a zero divisor, a unit, or none of the above. If it is a unit, give its inverse. If it is an zero divisor, show that this is the case. \vfill \item It turns out that there is a ring homomorphism $\varphi: \mathbb{Z}_{18} \to \mathbb{Z}_6$ which is onto and has $I=(6)=\mathrm{Ker}(\varphi)$. Is $\varphi$ one-to-one? Explain why or why not. What does the first isomorphism theorem say in this case? \vspace{1.25in} \item Is $I=(6)$ a prime or maximal ideal in $\mathbb{Z}_{18}$? Why or why not? \vspace{1in} \end{enumerate} \newpage \noindent {\large 2. (12 points)} Groups: Isomorphic or not. \begin{enumerate}[(a)] \item Explain why $A_4 \not\cong \mathbb{Z}_2 \oplus \mathbb{Z}_6$ [not isomorphic]. \vfill \item Explain why $Q = \{ \pm 1, \pm i, \pm j, \pm k \} \not\cong D_4$ [not isomorphic]. \vfill \item Explain why $U(10) \cong \mathbb{Z}_4$ [are isomorphic]. \vfill \end{enumerate} \noindent {\large 3. (8 points)} Rings: Explain why each pair of {\bf rings} are not isomorphic. \begin{enumerate}[(a)] \item $\mathbb{Z} \not\cong \mathbb{E}$ \qquad where \quad $\mathbb{E} = 2\mathbb{Z} = \{ n \in \mathbb{Z} \;|\; n \mbox{ is even}\}$. \vfill \item $\mathbb{R}^{3 \times 3} \not\cong \mathbb{C}$ \qquad where \quad $\mathbb{R}^{3 \times 3}$ is the ring of $3 \times 3$ real matrices \quad and \quad $\mathbb{C}$ is the complex numbers. \vfill \end{enumerate} \newpage \newpage \noindent {\large 4. (9 points)} Workin' in $\mathbb{Z}_{88}$. \qquad [Note: $88 = 2^3 \cdot 11$] \begin{enumerate}[(a)] \item Fill out the following table (for $\mathbb{Z}_{88}$): \begin{tabular}{|r|c|c|c|c|c|c|c|c|} \hline order = & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline \parbox[c][0.5in]{1.25in}{number of elements with this order = } & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace & \myspace \\ \hline \end{tabular} \item Draw $\mathbb{Z}_{88}$'s lattice of ideals. \vspace{2in} \item Which ideals are prime? maximal? \vspace{0.5in} \end{enumerate} \noindent {\large 5. (6 points)} Workin' in $\mathbb{Z}_{323}$. \begin{enumerate}[(a)] \item Is $221$ zero, a unit, a zero divisor, or none of the above in $\mathbb{Z}_{323}$? If $221$ is a zero divisor, prove it. If $221$ is a unit, find it's inverse. \vspace{2in} \item Is $20$ zero, a unit, a zero divisor, or none of the above in $\mathbb{Z}_{323}$? If $20$ is a zero divisor, prove it. If $20$ is a unit, find it's inverse. \end{enumerate} \newpage \noindent {\large 6. (18 points)} Sub-things \begin{enumerate}[(a)] \item Let $H = \{ 1, y \}$. Explain why $H$ is a subgroup of $D_5 = \{1,x,x^2,x^3,x^4,y,xy,x^2y,x^3y,x^4y\}$ (of course $x^5=1$, $y^2=1$, and $xyxy=1$), then show $H$ is {\bf not} a {\bf normal} subgroup of $D_5$. \vfill \item Let $\displaystyle S = 2\mathbb{Z} \oplus 3\mathbb{Z} = \left\{ (2x,3y) \;|\; x,y \in \mathbb{Z} \right\}$. Show $S$ is a {\bf subring} of $\mathbb{R} \oplus \mathbb{Q}$. \vfill \item Let $\mathbb{Q}[i] = \{ a+bi \;|\; a,b \in \mathbb{Q} \}$ where $i=\sqrt{-1}$. Show that $\mathbb{Q}[i]$ is a sub\underline{\bf field} of the complex numbers $\mathbb{C}$. \vfill \item Can a subring of an integral domain fail to be an integral domain? What about a quotient of an integral domain? \vfill \end{enumerate} \newpage \noindent {\large 7. (15 points)} An ideal question. \begin{enumerate}[(a)] \item Let $\varphi:R \to S$ be a ring homomorphism. Show that $\mathrm{Ker}(\varphi)$ is an ideal of $R$. \vfill \item Let $R$ be a commutative ring with $1$ and $I$ an ideal of $R$. Show that $I=R$ if and only if $I$ contains a unit of $R$. \vfill \item Let $\varphi:R \to S$ be a ring homomorphism where $\mathrm{Ker}(\varphi)=\{0\}$. Show that $\varphi$ is one-to-one. \vfill \end{enumerate} \newpage \noindent {\large 8. (8 points)} The Fundamental Theorem of Finite Abelian Groups. \begin{enumerate}[(a)] \item List all of the non-isomorphic abelian groups of order $100 = 2^2 \cdot 5^2$. Circle any that are cyclic. \vspace{1.5in} \item Which of the abelian groups of order 100 contain elements of order 25? \vspace{0.75in} \end{enumerate} \noindent {\large 9. (7 points)} Let $H = \{ (1), (12)(34), (13)(24), (14)(23) \}$. It can be shown that $H \triangleleft A_4$. Write down a Cayley table for $\diagquotient{A_4}{H}$. Is this quotient a cyclic group? \vfill \noindent {\large 10. (7 points)} Recall that if $G$ is a group and $g \in G$, then $\varphi_g(x)=gxg^{-1}$ is called an inner automorphism. Prove that $\varphi_g$ is an automorphism. \vfill \end{document}