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\begin{document}

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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 4.1  \\ Spring 2023 \hfill Due February 8}}}

\large

\vspace{1cm}

\begin{exercise}
Let $a, b \in \Z$.  Use the classification of the subgroups of $\Z$ to prove that
\[
a \Z \cap b \Z = \operatorname{lcm}(a,b) \Z.
\]
For $n \in \Z$ we are using the alternate (number theoretic) notation $n \Z = \langle n \rangle$ for the subgroup of $\Z$ generated by $n$.
\end{exercise}

\begin{exercise}
A group $G$ is called {\em finitely generated} if there exist $x_1, x_2, \ldots, x_n \in G$ so that $G = \langle x_1, x_2, \ldots, x_n \rangle$.  

\begin{itemize}

\item[\bf a.]  Prove that $\Z^n = \underbrace{\Z \oplus \Z \oplus \cdots \oplus \Z}_{n \text{ times}}$ can be generated by $n$ elements, and no fewer (this requires a little bit of linear algebra).

\item[\bf b.]  Prove that $\Q$ is {\em not} finitely generated.  [{\em Suggestion:}  Argue by contradiction.]

\end{itemize}

\end{exercise}

\begin{exercise}
Lang, Exercise II.1.5.  [{\em Warning}:  This is {\em not} simply the explicit form of $\langle S \rangle$ stated in class.]
\end{exercise}

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