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\title{Math 725\\Fall 2006\\Homework 5}
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\centerline{\it Due Wednesday, November 29, 2006.}

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\noindent 1. (5 pt) Let $F\subseteq K$ be fields. Show that $(F+K[x])[t]$ is an HFD if and only if $F$ is algebraically closed in $K$.

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\noindent 2. (5 pt) We have seen that for $R[x]$ to be an HFD, then $R$ must be an integrally closed HFD. Is it true that $R$ must be a completely integrally closed HFD (this is certainly true if $R$ is Noetherian)? Prove the statement or give a counterexample.

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\noindent 3. We say that the domain $R$ is an AP-domain if all atoms (irreducibles) in $R$ are prime.
\begin{itemize}
\item[a)] (5 pt) Show that any GCD-domain is an AP-domain.
\item[b)] (5 pt) Show that if $R$ is atomic then the following are equivalent:
\begin{enumerate}
\item $R$ is a GCD-domain.
\item $R$ is an AP-domain.
\item $R$ is a UFD.
\end{enumerate}
\item[c)] (5 pt) Show that if $R$ is a GCD-domain, then $R[x]$ is a GCD-domain.
\item[d)] (5 pt) Does c) hold for power series extensions?
\item[e)] (5 pt) Are the notions of GCD-domain and AP-domain equivalent?
\end{itemize}
 




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