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\title{Math 724\\Fall 2010\\Homework 4}
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\centerline{\it Due Monday, April 6, 2010}

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\noindent 1. (5 pt) Show that if $R$ is Noetherian and $\omega$ is almost integral over $R$ then $\omega$ is integral over $R$.

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\noindent 2. Let $F\subseteq K$ be fields.
\begin{itemize}
\item[a)] (5 pt) Show that $F+xK[x]$ is an HFD.
\item[b)] (5 pt) Show that $(F+xK[x])[t]$ is an HFD if and only if $F$ is algebraically closed in $K$.
\item[c)] (5 pt) It is known that if $R[x]$ is an HFD, then $R$ must be integrally closed. Explain why completely integrally closed is not necessary.
\end{itemize}

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\noindent 3. Give examples of domains with the following properties.
\begin{itemize}
\item[a)] (5 pt) An atomic domain with nonatomic integral closure.
\item[b)] (5 pt) A nonatomic domain (with at least one irreducible) where everything that can be factored into irreducibles, factors uniquely.
\item[c)] (5 pt) A domain (not a field) that contains no atoms whatsoever.
\item[d)] (5 pt) A nonatomic domain where every irreducible is prime.
\end{itemize}






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