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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 8.3  \\ Spring 2019 \hfill Due March 20}}}

\large

\vspace{1cm}

\begin{exercise}  Recall 
\[
\Z_{(p)} = \left\{  \left. \frac{r}{s}  \, \right| r \in \Z, s \in \N, p \nmid s \, \right\},
\]
which is a subgroup of $\Q$.

\begin{itemize}

\item[\bf a.]  Show that for any $k \in \N$,
\[
p^k \, \Z_{(p)} = \{ p^k a \, | \, a \in \Z_{(p)} \}
\]
is a subgroup of $\Z_{(p)}$.

\item[\bf b.]  Inclusion composed with the canonical epimorphism yields a homomorphism 
\[
f: \Z \to \Z_{(p)} / p^k \, \Z_{(p)}.
\]
Prove that $f$ is surjective and use the First Isomorphism Theorem to prove that
\[
\Z/p^k\Z \cong \Z_{(p)} / p^k \, \Z_{(p)}.
\]

\end{itemize}

\end{exercise}


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