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\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

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