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\name{}
\class{Math 171 - Abstract Algebra I Section \uline{\;\;}}
\assignment{HW \# 15}
\duedate{11/1/10}

\newcommand{\Z}{\mathbb{Z}}

\begin{document}

\problemlist{Dummit \& Foote (8.2) 3, 4, 5}

\begin{problem}[8.2.3]
Prove that a quotient of a P.I.D. by a prime ideal is again a P.I.D.
\end{problem}
\newpage

\begin{problem}[8.2.4]
Let $R$ be an integral domain.  Prove that if the following two conditions hold then $R$ is a Principal Ideal Domain:
\begin{itemize}
\item[(i)] any two nonzero elements $a$ and $b$ in $R$ have a greatest common divisor which can be written in the form $ra+sb$ for some $r, s \in R$, and 
\item[(ii)] if $a_1, a_2, a_3,\ldots$ are nonzero elements of $R$ such that $a_{i+1}|a_i$ for all $i$, then there is a positive integer $N$ such that $a_n$ is a unit times $a_N$ for all $n \geq N$.
\end{itemize}

\end{problem}
\newpage

\begin{problem}[8.2.5]
Let $R$ be the quadratic integer ring $\Z[\sqrt{-5}]$.  Define the ideals $I_2 = (2, 1+\sqrt{-5})$,  $I_3 = (3, 2+\sqrt{-5})$, and  $I_3' = (3, 2-\sqrt{-5})$.
\begin{itemize}
\item[(a)] Prove that $I_2$, $I_3$, and $I_3'$ are nonprincipal ideals in $R$. [Note that Example 2 following Proposition 1 proves this for $I_3$.]
\item[(b)] Prove that the product of two nonprincipal ideals can be principal by showing that $I_2^2$ is the principal ideal generated by 2, i.e., $I_2^2 = (2)$.
\item[(c)] Prove similarly that $I_2I_3 = (1-\sqrt{-5})$ and $I_2I_3' = (1+\sqrt{-5})$ are principal.  Conclude that the principal ideal $(6)$ is the product of 4 ideals: $(6) = I_2^2I_3I_3'$.
\end{itemize}

\end{problem}
\end{document}