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\begin{document}
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\begin{center}\textbf{Math 31 Take Home Final}\\Due August 28, 2018\end{center}

\textbf{Instructions:} This is an open notes, open book exam. You may use your notes or homework assignments from class, as well as Pinter 2nd edition. You may not use the internet or any other textbooks, nor may you confer with classmates. If you use any unauthorized aid, this is an honor code violation. This exam is due at 4pm on August 28.

\begin{enumerate}

\item Let $H$ and $K$ be normal subgroups of a group $G$ such that $H\cap K=\{e\}$. Prove that $G$ is isomorphic to a subgroup of $G/H\times G/K$.

\item 
\begin{enumerate}
\item Prove that any group of order $4$ is abelian.
\item Let $G$ be a group and $Z(G)$ be the center of $G$. Show that if $G/Z(G)$ is cyclic, then $G$ is abelian.
\item Let $[G:Z(G)]=4$. What is the isomorphism class of $G/Z(G)$?
\end{enumerate}

\item Let $G$ be an abelian group of order $16$ with elements $a,b\in G$ such that the order of both $a$ and $b$ is $4$, but $a^2\neq b^2$. What is the isomorphism class of $G$?

\item Show that $\Z$ is a principal ideal domain.

\item Let $I=\left<2\right>$ be an ideal of $\Z$. 
\begin{enumerate}
\item Is $I$ a maximal ideal of $\Z$? 
\item Is $I[x]$ an ideal of $\Z[x]$? Is it maximal?
\end{enumerate}

\item Suppose that $a$ and $b$ are elements of a commutative ring $R$ such that $ab$ is a zero-divisor. Show that either $a$ is a zero-divisor or $b$ is a zero-divisor.

\item Let $R$ be a ring with $m$ elements. Show that the characteristic of $R$ divides $m$.

\item Let $f(x)=x^3+6\in \Z_7[x]$. Write $f(x)$ as a product of irreducible polynomials.

\item Let $R$ be an integral domain. Prove that if every proper ideal is a prime ideal, then $R$ is a field. (\textit{Hint:} Consider the ideal $\left<a^2\right>$ for an element $a\in R$).
\end{enumerate}
\end{document}