%%% Template for MA 361 homework, after Eric J. Malm's hmcpset class
%
%%% PREAMBLE -----------------------------------------------------------
% Do make any changes before \begin{document}, except to add macros
%
\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{boxedminipage}
\usepackage{graphicx}
\usepackage[margin=1in]{geometry}
\usepackage{mathpazo}
\usepackage{fancyhdr}

\setlength{\headheight}{15pt}
\setlength{\parindent}{0pt}

% include this to have lists enumerated by letters
\renewcommand{\labelenumi}{(\alph{enumi})}
\renewcommand{\labelenumii}{\roman{enumii}.}
\renewcommand{\labelenumiii}{\Alph{enumiii}.}



\newcommand{\collaborators}[1]{\begin{center}\large\sffamily{I collaborated with #1}\end{center}}

\newenvironment{problem}[1][]{%
  \bigskip% put space before problem statement box %
  \noindent\begin{boxedminipage}{\columnwidth}%
  \def\@tempa{#1}%
  \ifx\@tempa\empty\else%
    \textbf{#1}\hspace{0.5em}%
  \fi%
}{%
  \end{boxedminipage}%
}



\pagestyle{fancyplain}
\lhead{\textbf{\NAME}}
\chead{\textbf{\COURSE: HW\ \HWNUM}}
\rhead{\textbf{\DUEDATE}}
\cfoot{}


%%% MACROS -----------------------------------------------------------
% You may add more macros to this list
%
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}

\newcommand{\GL}{\operatorname{GL}}
\newcommand{\Aut}{\operatorname{Aut}}




















\begin{document}
%%% NAME AND ASSIGNMENT NUMBER -----------------------------------------------------------
% Edit the values below to identify yourself and the assignment
%

\newcommand\NAME{Name:}  % your name
\newcommand\HWNUM{3}  % the homework number
\newcommand\DUEDATE{9/9/2019} % the due date
\newcommand\COURSE{MA 361}  % the course number

%%% COLLABORATORS -----------------------------------------------------------------------------------
% If you worked with other students on this assignment, acknowledge them here
% If you did not collaborate with anyone, comment out this line
%

%\collaborators{Ferdinand Georg Frobenius and Emmy Noether}


%%% PROBLEMS AND SOLUTIONS -------------------------------------------------------------------
% Include all problems statements
% For solutions, use the \begin{proof} environment
% Note that \begin{proof}[Solution] will replace the label "Proof" with "Solution", etc.
%

\begin{problem}[1]
\begin{enumerate}
\item Define a binary operation $*$ on $\QQ$ by
\[ a * b = a + b - ab. \]
Answer the following questions
\begin{enumerate}
\item Is $*$ associative?
\item Is $*$ commutative?
\item Is there an identity element for $*$?
\item Which elements of $\QQ$ have an inverse with respect to $*$?
\end{enumerate}

In each case, justify your answer with a brief proof or explanation.

\item Define a binary operation $\star$ on $\QQ$ by
\[ a \star b = \frac{ab}{3}. \]
Answer questions (i.)--(iv.) for the binary operation $\star$.
\end{enumerate}
\end{problem}

%\begin{proof}
%
%\end{proof}


\newpage


\begin{problem}[2]
Let
\[ H = \Bigg\{
\begin{bmatrix}
	a & -b \\
	b & a
\end{bmatrix}
\biggm\vert a, b \in \RR \Bigg\} \subset M_2(\RR). \]
\begin{enumerate}
\item Show that $H$ is closed under matrix addition and matrix multiplication.
\item Show that $\langle \CC, + \rangle$ is isomorphic to $\langle H, + \rangle$.
\item Show that $\langle \CC, \cdot \rangle$ is isomorphic to $\langle H, \cdot \rangle$.
\end{enumerate}
This shows that $H$ is a \textbf{matrix representation} of the complex numbers $\CC$.
\end{problem}

%\begin{proof}
%
%\end{proof}


\newpage


\begin{problem}[3]
Let $G = \{ z \in \CC \mid z^n = 1 \ \text{for some} \ n \in \ZZ^+ \}$.
\begin{enumerate}
\item Prove that $G$ is closed under multiplication.
\item Show that $G$ is \emph{not} closed under addition.
\item Prove that $G$ is a group under multiplication. This group is called the group of \textbf{roots of unity} in $\CC$.
\end{enumerate}
\end{problem}

%\begin{proof}
%
%\end{proof}



\end{document}
