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\title{Math 724\\Fall 2010\\Homework 3}
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\centerline{\it Due Monday, March 8, 2010}

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\noindent 1. Let $d<0$ be a squarefree integer and $R$ the ring of integers of the field $\mathbb{Q}(\sqrt{d})$.
\begin{itemize}
\item[a)] (5 pt) Show that if $R$ is a UFD, then $d$ must be prime (or -1).
\item[b)] (5 pt) Show that if $R$ is an HFD, then $d=-1, p,$ or $pq$ where $p$ and $q$ are distinct primes.
\item[c)] (5 pt) What is the status to the converse of the statements in parts a) and b)?
\end{itemize}

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\noindent 2. Let $R$ be an integral domain and $K$ a field containing $R$. We consider the domain $D:=R+xK[x]$
\begin{itemize}
\item[a)] (5 pt) Show that $D$ is atomic if and only if $R$ is a field.
\item[b)] (5 pt) Show that if $D$ is atomic, then $D$ is an HFD.
\end{itemize}

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\noindent 3. We have shown in class that the domain $\mathbb{Z}[\sqrt{-3}]$ is an HFD. 
\begin{itemize}
\item[a)] (5 pt) Find all nonprime irreducibles in $\mathbb{Z}[\sqrt{-3}]$.
\item[b)] (5 pt) What is the status of $\mathbb{Z}[\sqrt{-3}][x]$? Is it an HFD?
\end{itemize}

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\noindent 4. The domain $\mathbb{Z}[\sqrt{-61}]$ has class group isomorphic to $\mathbb{Z}/6\mathbb{Z}$. You may use this fact (along with the fact that every ideal class contains infinitely many primes) to answer the following questions.
\begin{itemize}
\item[a)] (5 pt) Find all possible ideal factorizations of an irreducible in $\mathbb{Z}[\sqrt{-61}]$ (in terms of primes from classes in the class group).
\item[b)] (5 pt) Use this information to construct (in terms of prime ideals) an element with irreducible factorizations of length 6 and length 2.
\item[c)] (5 pt) Find a concrete example of such an element.
\end{itemize}



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