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\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Daily Homework \#21}
\date{Assigned Friday, 27 October 2017; due Monday, 30 October 2017}

\maketitle

\vspace{1ex}

\begin{problab}{21.1} 
Let $R$ be a commutative ring such that $\mathfrak{a} = R \smallsetminus R^\times$ is an ideal.  Show that $\mathfrak{a}$ is the unique maximal ideal in $R$.
\end{problab}

\begin{problab}{21.2}
Let $R=\Z[\sqrt{-5}]=\{a+b\sqrt{-5} : a,b \in \Z\} \subseteq \C$ be the subring generated by $1$ and $\sqrt{-5}$.
Let $\fraka=(2,1+\sqrt{-5}) \subseteq R$, and let $\frakp \subseteq R$ be a prime ideal.  
\begin{enumalph}
\item Show that $\fraka$ is prime.
\item Show that $\frakp \cap \Z$ is a nonzero prime ideal.  
\item If $2 \not\in \frakp$, show that $\fraka_{(\frakp)} = R_{(\frakp)}$.
\item If $2 \in \frakp$, show that $\frakp=\fraka$ and that $\fraka_{(\frakp)} = (1+\sqrt{-5})R_{(\frakp)}$.
\item Conclude that $\fraka_{(\frakp)}$ is locally free.
\end{enumalph}
\end{problab}

\end{document}