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\lhead{Sebastian Casalaina}
\chead{Homework 1}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AUTHOR INFO %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

\author[Casalaina]{Sebastian Casalaina}
\address{University of Colorado, Department of Mathematics,  Campus Box 395,
Boulder, CO 80309}
\email{casa@math.colorado.edu}
\date{\today}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TITLE AND ABSTRACT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{Homework 1 \\ \ \\ Abstract Algebra 1 \\ \ \\MATH 3140}

\begin{abstract} 
This is Homework 1.  The problems are from Fraleigh \cite[\S0]{fraleigh}:
\begin{itemize}

\item HW1a Fraleigh Exercises 0: 2, 4, 10, 12, 14, 17.

\item HW1b Fraleigh Exercises 0: 18, 19, 28, 30, 31, 36. 

\end{itemize}
\end{abstract}



\maketitle


\tableofcontents 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% HOMEWORK ASSIGNMENT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newpage 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SECTION 0 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{Section 0}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EXERCISE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Exercise 0.2} Describe the set $\{m\in \mathbb Z \mid m^2=3\}$ by listing its elements.  



\begin{proof}[Solution]
Since there is no integer whose square is equal to $3$, we have that 
$$\boxed{\{m\in \mathbb Z \mid m^2=3\}=\{ \ \};}$$
i.e., the empty set $\emptyset$. 
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EXERCISE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Exercise 0.4} 
Describe the set $\{m\in \mathbb Z \mid m^2-m<115\}$ by listing its elements.  





\begin{proof}[Solution]

\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EXERCISE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Exercise 0.10} 






\begin{proof}[Solution]

\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EXERCISE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Exercise 0.12} 






\begin{proof}[Solution]

\end{proof}








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EXERCISE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Exercise 0.14} 








\begin{proof}[Solution]

\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EXERCISE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Exercise 0.17} 






\begin{proof}[Solution]

\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EXERCISE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Exercise 0.18} 






\begin{proof}[Solution]

\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EXERCISE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Exercise 0.19} 






\begin{proof}[Solution]

\end{proof}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EXERCISE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Exercise 0.28} 






\begin{proof}[Solution]

\end{proof}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EXERCISE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Exercise 0.30} 






\begin{proof}[Solution]

\end{proof}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EXERCISE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Exercise 0.31} 
 





\begin{proof}[Solution]

\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EXERCISE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{Exercise 0.36} 
 





\begin{proof}[Solution]

\end{proof}






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