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\begin{document}

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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 3.1  \\ Spring 2019 \hfill Due February 6}}}

\large

\vspace{1cm}

\begin{exercise}
Given a prime $p$, let
\[
\begin{split}
\Z_{(p)} &= \left\{ \left. \frac{r}{s} \, \right| \, r \in \Z, s \in \N, p \nmid s  \right\},\\
\Z[p^{-\infty}] &= \left\{ \left. \frac{r}{p^{\, t}} \, \right| \, r \in \Z, t \in \N_0  \right\}.
\end{split}
\]
Prove that $\Z_{(p)}$ and $\Z[p^{-\infty}]$ are (additive) subgroups of $\Q$.
\end{exercise}

\begin{exercise}
For $n \in \N$, let $\boldsymbol{\mu}_n = \{z \in \C \, | \, z^n = 1 \}$.  Prove that $\boldsymbol{\mu}_n$ is a (multiplicative) subgroup of $\C^{\times}$.
\end{exercise}

\begin{exercise}
Given a complex number $z = x+iy$ ($x,y \in \R$), define its {\em complex conjugate} to be $\overline{z} = x - iy$.  Prove the following properties of conjugation.
\begin{itemize}

\item[\bf a.]  For all $z, w \in \C$, $\overline{z+w} = \overline{z} + \overline{w}$ and $\overline{zw} = \overline{z} \, \overline{w}$.

\item[\bf b.]  For any $z \in \C$, $z \overline{z} = |z|^2$.

\item[\bf c.]  For any $z \in \C$, $z = \overline{z}$ if and only if $z \in \R$.

\end{itemize}

\noindent Use parts {\bf a} and {\bf b} to show that $|zw| = |z| |w|$ for all $z, w \in \C$.

\end{exercise}

\begin{exercise}
Let $G$ be a group and suppose that $\mathcal{F}$ is a family of subgroups of $G$.  Prove that
\[
J = \bigcap_{H \in \mathcal{F}} H
\]
is a subgroup of $G$.
\end{exercise}

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