\documentclass{article}
\usepackage{amsmath, amssymb,verbatim, amsthm, graphicx}

%exercise environment
\newcounter{exercise}
\setcounter{exercise}{0}
\newenvironment{exercise}{\addtocounter{exercise}{1} \noindent{\bf{Exercise \theexercise.}}}{\vspace{.5cm}}

\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\mat}[4]{\left( \begin{array}{cc} #1 & #2 \\ #3 & #4  \end{array} \right)}
\newcommand{\smat}[4]{\left( \begin{smallmatrix} #1 & #2 \\ #3 & #4  \end{smallmatrix} \right)}

\textwidth 6.5in
\textheight 9in
\oddsidemargin 0in
\evensidemargin 0in
\headsep -0.5in

\setlength{\parskip}{0.25cm}

\pagenumbering{gobble}

\begin{document}

\begin{center}
\includegraphics[width=0.5in]{../ltr-logo.jpg}
\end{center}

\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 3.3  \\ Spring 2023 \hfill Due February 1}}}

\large

\vspace{1cm}

\begin{exercise}
For $z = a+bi \in \C$, where $a,b \in \R$ and $i^2 = -1$, recall that we defined $\overline{z} = a-bi$ and $|z| = \sqrt{a^2+b^2}$.

\begin{itemize}

\item[\bf a.]  Show that for all $z, w \in \C$ one has $\overline{z+w} = \overline{z} + \overline{w}$ and $\overline{z  w} = \overline{z} \,\overline{w}$.

\item[\bf b.]  Use part {\bf a} and the fact that $|z|^2 = z \overline{z}$ to show that $|zw| = |z| \, |w|$ for all $z,w \in \C$.

\end{itemize}

\end{exercise}

\begin{exercise}
For $n \in \N$, let $\boldsymbol{\mu}_n = \{ z \in \C^{\times} \, | \, z^n = 1 \}$.  The elements of $\boldsymbol{\mu}_n$ are called the $n$th {\em roots of unity}.

\begin{itemize}

\item[\bf a.]  Prove that $\boldsymbol{\mu}_n$ is a subgroup of $S^1$.

\item[\bf b.]  {\em Euler's Formula} states that 
\[
e^{i \theta} = \cos(\theta) + i \sin(\theta)
\]
for any $\theta \in \R$.  Show that $z \in S^1$ if and only if $z = e^{i \theta}$ for some $\theta \in \R$.

\item[\bf c.] The addition formulae for sine and cosine imply that
\[
e^{i (\theta + \phi) } = e^{i \theta} e^{i \phi}
\]
for any $\theta, \phi \in \R$.  Use this fact and part {\bf b} to show that
\[
z \in \boldsymbol{\mu}_n \,\,\, \Longleftrightarrow \,\,\, z= e^{\frac{2 k \pi i }{n}}
\]
for some $k \in \Z$.  Conclude that $|\boldsymbol{\mu}_n| = n$.

\end{itemize}

\begin{exercise}
Prove that $Z(\operatorname{GL}_2(\R)) = \{aI \, | \, a \in \R^{\times} \}$.  [{\em Suggestion:}  First show that $\{aI \, | \, a \in \R^{\times} \}$ is a subgroup of the center.  To prove the opposite inclusion, choose any $A \in Z(\operatorname{GL}_2(\R))$ and commute it with the matrices $\smat{}{1}{1}{}$ and $\smat{1}{1}{}{1}$.]
\end{exercise}

\end{exercise}

\begin{exercise}
Let
\[
\operatorname{O}_2(\R) = \left\{ A \in \operatorname{GL}_2(\R) \, | \, AA^{t} = I  \right\},
\]
where $A^t$ denotes the transpose of $A$ (reflection of its entries across the diagonal).  Show that $\operatorname{O}_2(\R)$ is a subgroup of $\operatorname{GL}_2(\R)$.
\end{exercise}

\end{document}
