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\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#10} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Apr. $23^\mathrm{rd}$, 2021}

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%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

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Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force


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\begin{description}
\item[\#1] Let $R$ be a commutative ring with $1$ and let $a \in R$. Recall that $(a) = \{ ra \;|\; r \in R\}$ is the principal ideal generated by $a$. In a commutative ring with $1$, one says that $a,b \in R$ are {\bf associates} if there exists some unit $u$ (i.e., $u \in U(R)$) such that $a=ub$.

\begin{enumerate}[(a)]
\item What are the associates of $-1+3i$ in $\mathbb{Z}[i]$?  
\item What are the associates of $-1+3i$ in $\mathbb{C}$?
\item Let $R$ be an integral domain. Show that $(a)=(b)$ if and only if $a$ and $b$ are associates.
\end{enumerate}

\item[\#2] Let $I=\{ a+bi \;|\; a \text{ and } b \text{ are even integers}\}$. Prove that $I \triangleleft \mathbb{Z}[i]$.

\item[\#3] Let $R$ be a ring and let $I,J \triangleleft R$. One defines $I+J = \{x+y \;|\; x \in I \text{ and } y \in J\}$ to be the {\it sum} of $I$ and $J$. Similarly, one defines $IJ = \{ x_1y_1+\cdots+x_\ell y_\ell \;|\; \text{for some } \ell \geq 0 \text{ and } x_1,\dots,x_\ell \in I \text{ and } y_1,\dots,y_\ell \in J\}$ to be the {\it product} of $I$ and $J$. It can be shown that $I+J$,\  $IJ$, and $I \cap J$ are ideals of $R$.

\begin{enumerate}[(a)]
\item Prove that $I+J \triangleleft R$.
\item Consider $I=(12)$ and $J=(10)$ in $\mathbb{Z}$. Calculate $I+J$, $IJ$, and $I \cap J$. 
\item Since $\mathbb{Z}$ is a principal ideal domain, $(a)(b)=(c)$, $(a)+(b)=(d)$, and $(a) \cap (b)=(\ell)$ for some $c,d,\ell \in \mathbb{Z}$. Make a conjecture about how $a$ and $b$ (where $a$ and $b$ are non-zero) are related to $c$, $d$, and $\ell$. 

{\it Note:} You don't have to prove your conjectures. Just tell me how you think this ``ideal arithmetic'' works.
\end{enumerate}

\item[\#4] As a quick reminder, in $\mathbb{Z}$ and in $\mathbb{Z}_n$, we know that subgroup = normal subgroup = cyclic subgroup = subring = ideal = principal ideal.
\begin{enumerate}[(a)]
\item Let $R$ be a finite commutative ring with $1$. Explain why ideals are prime if and only if they are maximal.
\item Find all the ideals of $\mathbb{Z}_{24}$ and draw the corresponding lattice. Which ideals are prime? Which are maximal?
\item Determine which ideals in $\mathbb{Z}$ are prime and which are maximal. [Prove your assertions.]

{\it Note:} Don't forget to consider the trivial ideal: $\{0\}$.
\end{enumerate}

\item[RESUBMIT] Type up Homework \#9 Problem \#4 and its solution in \LaTeX. 

\fbox{\parbox{6.5in}{
Prove that $\mathbb{Q}[\sqrt{5}] = \{ a+b\sqrt{5} \;|\; a,b \in \mathbb{Q} \}$ is a field. 
}}

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{\it Notes:} Use a subring (of $\mathbb{R}$) test to show it is a ring. Use a ``conjugate trick'' to compute multiplicative inverses. Don't forget to mention the extra ``trivial'' bits about being a field: It is a {\it commutative} ring with {\it unity}. Specifically note $1=1+0\sqrt{5} \in \mathbb{Q}[\sqrt{5}]$ (and of course $1\not=0$). 

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     When typing this problem up, please write carefully: Restate the problem. Write in complete sentences.  
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